WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY by Christopher Edward .H.. eil Dissertation submitted to the Faculty of the Graduate School of The University of Maryland in partial fulfillment of the requirements for the degree of Doctor of Philosophy 1990 c, Vol. I Mary/qn4 LD 3i 3I Advisory Committee: !1rofessor John Benedetto, Chairman/A dvisor . M'l~d Visiting Professor Hans Feichtinger Professor John Horvath HG ii) Professor Raymond Johnson Professor Perinkulam Krishnaprasad t.E. V{)/. I Folio - = ,11' Jll??? :~::1 @ Copyright by j: _;.' ' ?. ' Christopher Edward Heil {J:,,! i I 1990 :11"' ABSTRACT Title of Dissertation: WIENER AMALGAM SPACES IN GENERALIZED HARMONIC ANALYSIS AND WAVELET THEORY Christopher Edward Heil, Doctor of Philosophy, 1990 Dissertation directed by: Professor John J. Benedetto, Department of Mathematics This thesis is divided into four parts. Part I, Introduction and Notation, describes the results contained in the thesis and their background. Part II, Wiener Amalgam Spaces, is an expository introduction to Feichtinger's gen- eral amalgam space theory, which is used in the remainder of the thesis to for- mulate and prove results. Part III, Generalized Harmonic Analysis, presents new results in that area. Part IV, Wavelet Theory, contains exposition and miscellaneous results on Gabor ( also known as Weyl-Heisenberg) wavelets. Amalgam, or mixed-norm, spaces are Banach spaces of functions deter- mined by a norm which distinguishes between local and global properties of functions. Specific cases were introduced by Wiener. Feichtinger has devel- oped a far-reaching generalization of amalgam spaces, which allows general function spaces norms as local or global components. We use Feichtinger's amalgam theory, on d-dimensional Euclidean space under componentwise multiplication, to prove that the Wiener transform (introduced by Wiener to analyze the spectra of infinite-energy signals) is an invertible mapping of -- ________ ... ......._ .,... .....,.~~~~- the amalgam space with local L2 and global LtJ. components onto an appropri- ate space defined in terms of the variation of functions, for each q between one and infinity. As corollaries, we obtain results of Beurling on the Fourier trans- form and results of Lau and Chen on the Wiener transform. Moreover, our results are carried out in higher dimensions. In addition, we prove that the higher-dimensional variation spaces are complete by using Masani's helices; this generalizes a one-dimensional result of Lau and Chen. ,,J In wavelet theory, we present a survey of frames in Hilbert and Banach ~ :~I (. spaces and the use of the Zak transform in analyzing Gabor wavelets. Frames .? 'l are an alternative to unconditional bases in these spaces; like bases, they Provide representations of each element of the space in terms of the frame elements, and do so in a way in which the scalars in the representation are explicitly known. However, unlike bases, the representations need not be unique. We then discuss the specific case of Gabor frames in the space of square-integrable functions, concentrating on the role of the Zak transform in the analysis of such frames. ACKNOWLEDGEMENT It is a pleasure to thank the individuals who influenced this research. Foremost among these is my thesis advisor, John Benedetto, whose encour- agement, mathematical support, and previous mathematical results were in- valuable. I also thank my good friend David Walnut, now at Yale University, with whom I learned, and wrote joint papers on, wavelet theory, and who put up with me during three weeks of conferences and travel in Italy. This work would have been impossible without the mathematical frame- work of the Wiener amalgam spaces, provided by Hans Feichtinger of the University of Vienna. I thank Dr. Feichtinger for numerous preprints, discus- sions, suggestions, and encouragement. Also critical was the work of Benedetto with George Benke and Ward Evans of The MITRE Corporation on the higher-dimensional Wiener-Plan- cherel formula. I thank Dr. Benke, my group leader at MITRE, for suggesting and supporting my non-thesis work on wavelets, and Dr. Evans for numerous mathematical discussions and advice. Finally, my wife Sunnie deserves uncountably many more thanks than I can express here for her support and understanding throughout this endeavor. Without her, this work could never have been accomplished. ii TABLE OF CONTENTS CHAPTER PAGE PART I. INTRODUCTION AND NOTATION. 1 Chapter 0. Introduction. 2 Section 0.1. Amalgam spaces. 3 Section 0.2. Generalized harmonic analysis. 9 Section 0.3. Wavelet theory. 20 Chapter 1. Notation and Definitions. 28 Section 1.1. Basic symbols. 28 Section 1.2. Special sets. 30 Section 1.3. Functions. 32 Section 1.4. Convergence. 35 Section 1.5. Operators. 37 Section 1.6. Topological groups. 39 Section 1. 7. Function spaces. 41 Section 1.8. The Fourier transform. 45 Section 1.9. Group representations. 46 iii PART II. WIENER AMALGAM SPACES. 48 Chapter 2. Wiener amalgam spaces. 49 Section 2.1. Moderate weights. 52 Section 2.2. Definition and basic properties. 62 Section 2.3. Inclusion relations. 70 Section 2.4. Discrete norms. 76 Section 2.5. Duality. 87 PART III. GENERALIZED HARMONIC ANALYSIS. 92 Chapter 3. Besicovitch spaces. 93 Section 3.1. Rectangular limits. 95 Section 3.2. Equivalence with Wiener amalgam spaces. 104 Section 3.3. Beurling's characterization of B(p, oo ). 118 Section 3.3. A characterization of B(p, q). 137 Section 3.5. Weighted Besicovitch spaces. 144 Chapter 4. The Wiener transform. 147 Section 4.1. Definitions. 149 Section 4.2. The symmetric difference operator. 1.58 Section 4.3. The variation spaces. 167 Section 4.4. Continuity of the Wiener transform. 170 Section 4.5. lnvertability of the Wiener transform. 185 lV -- Chapter 5. Completeness of the variation spaces. 195 Section 5.1. Helices. 196 Section 5.2. Completeness. 206 PART JV. WAVELET THEORY. 218 Chapter 6. Frames. 219 Section 6.1. Bases. 221 Section 6.2. Frames in Hilbert spaces. 225 Section 6.3. Frames and bases. 238 Section 6.4. Atoms in Hilbert spaces. 242 Section 6.5. Frames and atoms in Banach spaces. 248 Section 6.6. Stability of atoms. 252 Chapter 7. Gabor systems and the Zak transform. 255 Section 7 .1. Gabor systems. 257 Section 7 .2. The Zak transform. 262 Section 7 .3. Gabor systems and the Zak transform. 268 Section 7.4. The Zak transform on LP(Rd). 275 Section 7 .5. Amalgam spaces and the Zak transform. 280 Section 7.6. Multiplicative completion. 283 References 290 V PART I INTRODUCTION AND NOTATION 1 CHAPTER 0 INTRODUCTION This thesis falls naturally into several parts. Part I, Introduction and Notation, describes the results contained in this thesis and their background, and lays out the notational scheme used through- out. Part I consists of Chapters O and 1. Part II, Wiener Amalgam Spaces, is an expository introduction to Fe- ichtinger's general amalgam space theory, which is used in the remainder of the thesis to formulate and prove results. Part II consists of a single chapter, Chapter 2. Part III, Generalized Harmonic Analysis, contains new results in that area. The results depend heavily on the use of amalgam spaces. Our major result links and extends results of Wiener, Beurling, Lau and Chen, and Benedetto, Benke, and Evans into a single isomorphism theorem. Part III consists of Chapters 3 through 5. Finally, Part IV, Wavelet Theory, contains exposition and miscellaneous new results in that area. Part IV consists of Chapters 6 and 7. We introduce each of Parts II, III, and IV below, in Sections 0.1, 0.2, and 0.3, respectively. 2 Section 0.1. Amalgam spaces. The classical LP spaces on the real line R consist of those functions f for which the norm 00 )1/p IIJIIP = (1-= lf(t)IP dt is finite. These spaces play a prominent role in modern analysis, yet often are difficult to use in applications because the LP norm does not distinguish between local and global properties. For example, all rearrangements of a given function have identical LP norms. Thus, it is not possible to recognize from the norm of a function whether it is, say, the characteristic function of an interval or the sum of many characteristic functions of small intervals spread widely over R. As another example, "local" and "global" inclusions in LP behave differently, with the result that there are no inclusion relations for LP as a whole. To illustrate this, let K C R be a compact set, and let f_P be the space of sequences {ck} which are p-summable, i.e., ~ icklP < oo. Define the following subspaces of LP(R): LP(K) - {f E LP(R) : supp(!) CK}, where X[k,k+I] is the characteristic function of the interval [k, k+ 1]. Functions in LP(K) have only "local" behavior, while functions in GP have only "global" behavior, in some sense. "Local" inclusions behave as follows: 3 while "global" inclusions behave as: No LP(R) is contained in any other Lq(R). Amalgam spaces decouple the connection between local and global proper- ties which is inherent in the definition of LP spaces. Their first use was by Norbert Wiener, in the formulation of his generalized harmonic analysis. In the notation of this thesis, he defined the spaces W(L1, L 2 ) and W(L 2 , L 1 ) in (W4], and W(L 1 ,L00 00 1) and W(L ,L ) in (Wl; W2], where W(LP,Lq) is the standard amalgam space defined by the norm ( n+I ) q/p) 1/q (0.1.1) 11/llw(LP,Lv) = ~ (1n lf(t)jP dt , the usual adjustments being made if p or q is infinity. Amalgams have been reinvented many times in the literature; the first systematic study appears to have been undertaken by Holland in (Ho]; an excellent review article is (FS]. The amalgams W(LP, Lq) distinguish between local LP and global Lq prop- erties of functions in the ways we expect. For example, rearrangements do not have identical norms in general, and inclusions behave correctly: The dual space of W(LP,Lq) is W(LP' ,Lq'), where lp +-.p! , = lq +-.q! , = 1. For 1 ~ p, q ~ 2 we have a Hausdorff-Young property for the Fourier transform: 4 a note that local and global properties are interchanged on the Fourier trans- form side. H. Feichtinger recently proposed a far-reaching generalization of amalgam spaces to general topological groups and general local/global function spaces, e.g., [F2; F8], cf., Chapter 2. Given Banach spaces B, C of functions on a locally compact group G, he defines spaces W(B, C) of functions or distribu- tions which are "locally in B" and "globally in C". Moreover, his generaliza- tion is powerful and natural. Some properties which follow immediately from his theory are the following. Inclusions. If B1 C B2 and C1 C C2 then W(B1,C1) C W(B2,C2). Duality. If a space of test functions (e.g., the Schwartz space S(R) of smooth, rapidly decreasing functions) is dense in Band C then W(B, C)' = W(B',C'). Complex interpolation. Complex interpolation can be carried out in each component of W(B, C) separately. Pointwise multiplications. If B1 ?B2 C B3 and C1 -C2 C C3 then W(B1, C1 )? W(B2, C2) C W(B3, C3). Convolutions. If B1 * B2 C B3 and C1 * C2 C C3 then W(B1, C1) * W(B2, C2) C W(B3, C3). Many other specific results follow immediately from Feichtinger's theory by choosing Sobolev spaces, Besov spaces, weighted LP spaces, the Fourier algebra A, etc., as the local or global components, with various choices of 5 topological groups. Feichtinger refers to his spaces W(B, C) as Wiener-type spaces; follow- ing a suggestion of J. Benedetto, and in order to promote the link between Feichtinger's generalization and the amalgams previously defined in the lit- erature, we call them Wiener amalgam spaces. Taking G to be the group R under addition with Haar measure dt, the local component B to be LP(R), and the global component C to be Lq(R), results in a Wiener amalgam space coinciding precisely with the standard amalgam space defined by (0.1.1). In this thesis we obtain new results, and new proofs and generalizations of previously known results, in generalized harmonic analysis (Part III) and in wavelet theory (Part IV), by using amalgam spaces. Except for the amalgam space connection, the results in the two parts are unrelated, although we believe that the application of wavelets to generalized harmonic analysis could produce new results in the future. For the benefit of the reader, we present in Part II a self-contained intro- duction to Feichtinger's theory. Since his theory is not needed in the later parts in its full generality, we present a simplified theory in which we allow only weighted LP spaces as local or global components. This results in a considerable technical simplification of the proofs without destroying their essential flavor. Thus, Part II can be considered an elementary introduc- tion to the general theory as presented in [F8]. In addition, we prove only those results directly related to our needs in this thesis, e.g., completeness, 6 translation invariance, equivalence of discrete norms, inclusions, and duality. Part II is purely expository, and it is not necessary to read Part II in order to appreciate the results in Parts III and IV. While Part II is written in terms of general topological groups, the results in Parts III and IV use the Wiener amalgam spaces on two specific topological groups. Part III uses the multiplicative group R~ = {x E Rd : Xj f=. 0, all j}, under componentwise multiplication, with Haar measure dt/lt 1 ? ? ? tdl? Part IV uses the additive group Rd, under component wise addition, with Haar mea- sure dt. To clearly distinguish between amalgam spaces on these two groups, we use the following notation in Parts III and IV ( and in Sections 0.2 and 0.3 of this chapter): (0.1.2) and (0.1.3) The amalgam space W(LP,Lq) on the group Rd is precisely the higher- dimensional analogue of the standard amalgam space defined in (0.1.1); the intervals [n,n+l] are simply replaced by cubes [n 1 ,n1 +1] x ??? x [nd,nd+l] for n E zd. We point out, however, that this norm is only equivalent to the fundamental norms used by Feichtinger as the basic definition for W(LP, Lq). We refer to a norm such as (0.1.1) as a discrete-type norm for W(LP, Lq); the fundamental defining norm is instead a continuous-type norm (Definition 7 2.2.2). Such norms more clearly illustrate the local LP /global Lq features of For the one-dimensional case ( d = 1), the discrete-type norm for W.(LP, Lq) on the group R. is (0.1.4) llfllw.(LP,L9 ) = ( L nEZ,? (1 dt)q/p)l/q IJ(t)l -ltl ? ?[2",2a+1] The higher-dimensional version of this norm is obtained by replacing the using the Haar measure dt/lt1 ? ? ? tdl? Special Acknowledgement. We thank Dr. Feichtinger for permission to use several of his unpublished lecture notes in this section. 8 Section 0.2. Generalized harmonic analysis. In this section we summarize and present background for results obtained in Part III of this thesis. Items a-d below discuss the background of our problems in generalized harmonic analysis, e-f discuss our results, g discusses future research possibilities, and h outlines Part III by chapters. a. The Wiener-Plancherel formula. The Fourier transform provides the basic definition of spectrum for finite-energy functions on the real line. Central to its definition is the Plancherel formula 1-: l/(t)l 2 dt = 1_: lib)l2 d-y, where the Fourier transform is defined by cf., Section 1.8. In order to deal with infinite-energy but finite-power func- tions, Wiener introduced what we now call the Wiener transform, and proved the Wiener-Plancherel formula, e.g., (Wl]. These are defined as follows. Given a function f on the real line R, its Wiener transform is (formally) oo e-21"i-yt _ X (t) (0.2.1) W f(-y) = J f(t) _[-i,iJ dt. -oo -21rit If f has bounded quadratic means, i.e., if (0.2.2) sup lT 1T lf(t)l 2 dt < oo, T>O 2 -T 9 then W f is well-defined (Theorem 4.1. 7). The Wiener-Plancherel formula states that for such f, 00 (0.2.3) lim - 1 1T lf(t)l 2 dt = lim -21 16..\Wf(,)12 d,, T-oo 2T -T .\-,Q ). -oo meaning that if one limit exists then the other does also and they are equal, and where 6..\ is the symmetric difference operator Note that if f has finite energy, i.e ., if f E L2 (R), then the left-hand side of (0.2.3) is zero. Wiener called the theory associated with (0.2.1) and (0.2.3) generalized harmonic analysis as it generalizes the usual finite-energy harmonic analysis. For background, perspective, and proof of (0.2.3) and associated subjects, see [B7]. The Wiener-Plancherel formula has been extended to higher dimensions in [BBE], [Bl], and [Ben]. The paper [BBE] adopted a "rectangular" ap- proach to higher dimensions, while [Bl] and [Ben] adopted a "spherical" approach. We prove our results in generalized harmonic analysis in higher dimensions following the rectangular approach of [BBE]. For clarity, we con- centrate in this introduction on one-dimensional statements, and summarize higher-dimensional results in item f below. b. Lau 's extension of the Wiener-Plancherel formula . K.-S. Lau and J. K. Lee observed in [LL] that the space of functions f for which the limit 10 on the left-hand side of (0.2.3) exists is nonlinear, and, more generally, that (0.2.4) B(p, lim) = {f E Lf c(R) : lim 2._ JT lf(t)jP dt exists} 0 T-+oo 2T -T is nonlinear (the case p = 2 had originally been proved in [HW]). Therefore B(p,lim) cannot be dealt with using the methods of ordinary functional anal- ysis. However, the Wiener transform W is defined for all f with bounded quadratic means, hence for all f E B(2,limsup), where B(p,limsup) is the space of functions f for which the norm 1 1T )1/p (0.2.5) IIJIIB(p,limsup) = li~_:;P ( 2T -T lf(t)IP dt is finite. Marcinkiewicz, in [Mar], proved that B(p, lim sup) is a Banach space once functions f, g E B(p,limsup) with II/ - YIIB(p,limsup) = 0 are identified. Lau and Lee proved that the Wiener transform W is a topological isomorphism of B(2,limsup) onto the space V(2,limsup), where = (2100 )l /p (0.2.6) IIFllv(p,limsup) limsup , jA,xF(--y)jP d--y ? >.--o I\ -oo Since V(p,limsup) is not solid, i.e., IFI ::; IGI does not necessarily im- ply l!Fllv(p,limsup) ::; IIGllv(p,limsup), the completeness of V(p,limsup) is a difficult question. Using the helix techniques of Masani, Lau and Lee were able to prove that V(p,limsup) is a Banach space (once functions F, G with IIF - Gllv(2,limsup) = 0 are identified), cf., [LL; Ml; M3]. Following Lau and Lee's work on B(p, lim sup), Lau and Chen proved in (CLl] that the Wiener transform W extends to a topological isomorphism of 11 the space B(2, oo ), where (0.2.7) llf llB(p,oo) onto V(2, oo ), where (0.2.8) IIFllv(p,oo) = (2100 )1/p sup >." l~~Wf(-y)jPd-y . ~>O -oo We reproduce the proof of this result in Section 4.4-4.5. Our results include and generalize this result, both to a larger class of spaces and to higher di- mens1ons. It is clear that B(p, oo) is a Banach space, without the need to form equiv- alence classes other than the usual a.e. ones. Lau and Chen proved that V(p, oo) is also a Banach space (after the formation of equivalence classes), by using Masani 's helix techniques. c. Beurling's AP and BP spaces. In one of his deep investigations into spectral synthesis, Beurling introduced the following spaces, e.g., [Bel]: (0.2.9) BP - n Li(R) wEA. and (0.2.10) AP' u Lwp' ,(R), wEA. where A is the class of even, positive, integrable weights which are decreasing on (0,oo), i + J, = 1, t 1-p' W - w ' 12 and L{:, is defined by the norm 00 ) 1/p IJJIILt = (1_ lf(t)IP w(t) dt . 00 Beurling proved the following facts. AP and BP are Banach spaces. AP C L 1 (R) and is a convolution algebra. (AP)' = BP', under the duality (0.2.11) (f, g) - 1-: f(t) g(t) dt. BP= B(p,oo). In addition, he proved that the Fourier transform on A2 satisfies an isomor- phism property similar to the one proved by Lau and Chen for the Wiener transform on B(2, oo) = B 2 ? Recasting his result into our terminology, he essentially proved that the Fourier transform is a topological isomorphism of A2 onto a space V(2, 1) defined by the norm (0.2.12) IIFllvc2,1) Jo1 = (21-oo l~.xF("Y)l2 )1/2 Td>.= :X- _ d1 ? 00 The proof required tricky estimates involving the weights w; we reproduce it in Section 4.4. Many of Beurling's results in [Bel) (with the exception of the Fourier transform isomorphism theorem) were actually proved in higher dimensions, 13 but with a spherical approach, rather than the rectangular approach of this thesis. d. Feichtinger's contribution. As discussed in Section 0.1, Feichtinger has produced a general theory of amalgam spaces on topological groups. In [F4], he characterized B(p, oo) as an amalgam space by proving that (0.2.13) B(p,oo) = W.(V,L 00 ) = W(LP(R.),L 00 (R.)), under equivalent norms. This insight provided us with a framework to link Beurling's and Lau's isomorphism results, and to prove our own results. The characterization as an amalgam space provided us with equivalent discrete- type norms, which are the basic machinery we use to prove our major thee- rems. e. Our results. For clarity, we discuss one-dimensional versions of our results first, and make remarks on the higher-dimensional formulations in item f. We generalize Feichtinger's characterization of B(p, oo) as the amalgam space W.(LP, L 00 ) as follows. Define B(p, q) to be the space of functions f for which the norm 00 / ( 1 JT )q/p dT)l/q (0.2.14) 11/IIB(p,q) = ( Jo 2T -T lf(t)IP dt T is finite, with the standard adjustments if p or q is infinity. In Theorem 3.2.4 we prove that (0.2.15) 14 with equivalent norms. This provides us with discrete-type norms for all B(p, q), cf., (0.1.4). I Recall now that (AP)' = BP , with duality defined by (0.2.11). From 1 (0.2.13), we have BP' = B(p',oo) = W.(LP ,L00 ). It follows immediately from Feichtinger's amalgam theory that W.(LP', L00 ). However, these amalgam spaces are on the multiplicative group R., so the duality is with respect to the Haar measure on R., i.e., with I - dt (f,g) = la.t(t )g(t) itT I ! .:I, It therefore follows that .. i.e., / E AP ?:> tf(t) E w.(LP, L1 ) = B(p, 1). Except for the convergence factor X[-l,l](t), the Fourier transform off E A2 therefore corresponds to the Wiener transform of tf(t) E B(2, 1), i.e., ~ -271"i W(tf)(--y). 15 Since the convergence factor is not needed to make the integral defining W g converge for g E B(2,l), and is irrelevant once we compute A>.Wg, the Beurling isomorphism theorem for the Fourier transform on A2 therefore im- plies that the Wiener transform is a topological isomorphism of B(2, 1) onto V(2, 1). Campa.ring this to the Lau result, that Wis a topological isomor- phism of B(2,oo) onto V(2,oo), we anticipate the major result of Pa.rt III, namely, that W is a topological isomorphism of B(2, q) onto V(2, q) for each 1 $; q < oo (Theorem 4.5.5), where V(p, q) is defined by the norm 00 00 (0.2.16) = ( (2 ! )q/p Td> .)1/q IIFllv(p,q) ( Jo >. _ 16.>.F("Y)IPd'Y . 00 We prove our isomorphism theorem directly, without interpolation. This avoids lengthly technical details establishing the interpolation properties of the non-solid spaces V(p, q). Moreover, our use of Wiener amalgam spaces to prove this result gives new proofs of the Beurling and Lau results using a single technique, rather than the very different techniques used by the original authors. Although not needed to prove our isomorphism theorem, we show in Sec- tion 3.4 that B(p, q) can be written as a union or intersection of weighted V spaces, simila.r to the Beurling characterizations of AP, BP given in (0.2.9) and (0.2.10), cf., Proposition 3.4.6. This characterization allows us to relate the spaces B(p, q) to other spaces which have appeared in ha.rmonic analysis, cf., Remark 3.4.7. f. Higher dimensions. Benedetto, Benke, and Evans, in [BBE), extended 16 the Wiener-Plancherel formula (0.2.3) to higher dimensions, in a "rectangu- lar" way. This nontrivial task included the determination of correct higher- dimensional analogues oflimits, the Wiener transform W, and the symmetric difference operators ~>., as well as the formulation and proof of new Taube- rian theorems. The term "rectangular" stems from the fact that the intervals [-T, T] in (0.2.3) are replaced by rectangular boxes RT= IT1=1[-T;, T;] for T = (T1, ... , Td) E Ri. For example, the space B(p, oo) is defined in higher dimensions in a rectangular way by the norm (0.2.17) 11/IIB(p,oo) = sup ( ; I / 1/(t)IP dt) l/p. TERt 1 T }RT The rectangular higher-dimensional definitions of limits are given in Sec- tion 3.1, of the Wiener transform in Section 4.1, and of the difference op- erator in Section 4.2. Using those definitions, the Wiener-Plancherel formula becomes the following: for/ E B(2, oo ), (0.2.18) lim IRl I r lf(t)12 dt = lim l>i 2d A I / ,~>.WJ(-y)l2 d,y. T-+oo T JR T >.--o 1 ? ? ? d ]a,t1. We prove all our results in higher dimensions using the higher-dimensional rectangular definitions. This includes the characterization of B(p, q) as an amalgam space, the convergence of the Wiener transform on B(2, q), the isomorphic nature of the Wiener transform as a mapping of B(2, q) onto V(2, q), and the proof of the completeness of the higher-dimensional variation spaces V(p, q). The completeness of V(p, q) is proved in the final chapter of Part III. For one dimension, the completeness follows as a corollary of results of Lau and Chen 17 based on Masani's helix techniques. In higher dimensions, the proof requires an iteration of those techniques (Theorem 5.2.3). We review the definitions and basic properties of helices in that chapter, and, while not appropriate for proving the completeness of V (p, q), we also indicate how to extend helices directly to Rd. g. Future results. Benedetto has completed, and Benke is completing, work on spherical higher-dimensional analogues of the Wiener-Plancherel formula, cf., [Bl] and [Ben], spherical in the sense that the intervals [-T, T] in (0.2.3) are replaced by spheres of radius T. The resulting spherical formulas appear to be even more interesting than their rectangular counterparts. A major goal for future research is therefore to determine the spherical analogues of our isomorphism theorems. Another goal is to investigate higher-dimensional analogues of the Lau and Lee isomorphism theorem on B(2,limsup), both in rectangular and spherical settings. A related area in which we expect our amalgam space methods to be of use is the following. In [CLl], Lau and Chen proved modified Wiener-Plancherel isomorphism theorems, obtained by replacing the factors 1/2T by 1/(2Tt. Such results have applications to fractals, Hausdorff measures, etc., cf., [E2; Stl; St2]. A goal for future research is therefore to prove our isomorphism theorem in such a setting. As a step in this direction, we prove in Section 3.5 that the spaces Bp(p, q), obtained by replacing the factors 1/IRTI in the def- inition of the higher-dimensional B(p, q) by general functions p(T), can be 18 written as weighted Wiener amalgam spaces on the multiplicative group. h. Outline. We outline Part III by chapters. In Chapter 3 we present the definitions and fundamental characterizations of the Besicovitch spaces B(p, q). We prove that B(p, q) coincides with the Wiener amalgam space W..,(LP, Lq) and prove bounds for the norm equiv- alence. We discuss the relationship of B(p, q) to unions or intersections of weighted LP spaces. We discuss the effect of replacing the factor 1/2T in the definition of B(p, q) (or 1/IRTI in higher dimensions) by a general function p(T), and show that the resulting spaces are again Wiener amalgam spaces, with weighted LP components. . ! In Chapter 4 we discuss the Wiener transform. We prove that it is defined ~ on each space B(2, q) for 1 :5 q :5 oo, and determine the basic properties ,...i. of A.\ W f. We reproduce the Beurling and Lau proofs of the isomorphic nature of W on B(2, 1) and B(2, oo ), respectively, and then prove, directly, the continuity and invertibility of W on each of the spaces B(2, q) by using the Wiener amalgam norms derived in Chapter 3. In Chapter 5 we prove that the variation spaces V(p, q) are Banach spaces by using an adaptation of Masani 's helix techniques. We review the basic def- initions and properties of helices and give Lau and Chen's proof that V(p, oo) is complete when d = 1, then extend this proof to higher dimensions by using an iterated helix technique. 19 Section 0.3. Wavelet theory. Part IV of this thesis is a survey of results in wavelet theory, especially frames, Gabor systems, and the Zak transform. Part IV is largely expository; results of many authors have been combined with examples, remarks, and minor results of our own into a survey of one portion of wavelet theory. Most of the work on Part IV was completed prior to 1988, when we were hired by The MITRE Corporation to pursue work in wavelets. After that point we concentrated our thesis work on generalized harmonic analysis. Our work on wavelets for MITRE has appeared under separate cover, e.g., [BHW; Hl; H2; HWl; HW2]. The paper [HW2] is a comprehensive introduction to . J wavelet theory from the point of view of frames. ,.i. . In item a below we discuss the basic problem of wavelet theory. Item b discusses frames, which are an alternative to orthonormal or unconditional bases. Items c and d discuss Gabor and affine wavelets, respectively, and item e discusses the general wavelet theory of Feichtinger and Grochenig. Item f outlines Part IV by chapters. a. Wavelet theory. The basic problem of wavelet theory is to find good bases, or good substitutes for bases, for Banach function spaces, especially L2 (Rd), the Hilbert space of square-integrable functions on d-dimensional Euclidean space. The term "good" has, of course, many interpretations, in- eluding, but not limited to, the following. The basis elements should be easily generated from a single ( or finitely many) functions through a combination 20 of the fundamental operations of translation, modulation (translation in fre- quency, i.e., multiplication by e21?+r?t), and dilation. The basis elements should be well localized in time and frequency, i.e., both the basis elements and their Fourier transforms should have good decay. Both the basis elements and their Fourier transforms should be smooth, preferably infinitely differentiable. Two basic approaches to constructing such systems have developed. These are the Gabor ( or Weyl-Heisenberg) wavelet systems and the affine wavelet systems, discussed below in items c and d. We point out that it has recently become unfashionable to refer to Gabor systems as wavelets, the term wavelet instead being reserved for affine systems. b. Frames. Frames were invented by Duffin and Schaeffer, in the course of an investigation into nonharmonic Fourier series, as an alternative to or- thonormal bases in Hilbert spaces [DS]. A sequence { en} of vectors in a Hilbert space H is an orthonormal basis if the sequence is orthonormal, i.e., (em, en) = 0 if m-=/:- n and (en, en) = 1, and the Plancherel formula holds, i.e., r: l(x, en)l 2 = llxll 2 for all x EH. It follows that if x E H then there exist unique scalars {en} such that x = r: enen, A sequence {xn} in H is a frame if there exist numbers A, B > 0 such that A llxll 2 :::; r: l(x, en)l 2 2 :::; B llxll for x E H. The vectors {xn} need not be orthogonal, yet it follows that given x E H there exist scalars {en} such that x = r: enXn? Unlike orthonormal bases, these scalars need not be unique. However, they are given explicitly, and the series x = I: enXn converges 21 unconditionally, i.e., all rearrangements converge (and converge to x ), cf., Proposition 6.2.8. Frames which are ezact, i.e., for which the representations x = ~ CnXn are unique, are bounded unconditional bases for the Hilbert space, and vice versa (Proposition 6.3.3). Frames thus provide representations of elements of a Hilbert space in terms of the frame elements, like orthonormal bases. Since the definition of frame is less restrictive than the definition of orthonormal bases, frames are usually easier to construct in applications. c. Gabor systems. A Gabor system is generated from a single function ( the mother wavelet) by translations and modulations; in particular, a Gabor system for L2 (R) has the form {9mn}m,nEZ, where 9mn(t) = e2 ff'imbt g(t - na), and g E L 2 (R) and a, b > 0 are fixed. Gabor systems have a long history and are closely related to several well-known signal processing tools, e.g., the short- time Fourier transform, the Wigner distribution, and the radar ambiguity function, cf., [DeJ]. They have applications to many areas, e.g., quantum mechanics [BZ; BZZ; Zl; Z2; Z3] and holography and optical computing [Sehl; Sch2; Sch3]. We restrict our discussion here to one dimension; the extension to higher dimensions is essentially trivial. We concentrate in this thesis on the case of Gabor systems satisfying ab = 1. This case is especially amenable to analysis through the use of the Zak trans- form, a tool which has been reinvented many times in the literature. Accord- 22 ing to Schempp, a discrete form of the Zak transform was used by Gauss. Janssen, in [Jl], lists some of the other occurances of the Zak transform. Zak used the transform in quantum mechanics to study the Gabor system gener- 2 ated by the Gaussian function g( t) = e-71't ? Some of the earliest results on the Zak transform were obtained by Auslander and Tolimieri by topological methods, e.g., [AT2], cf., [ATl; AGT; AGTE]. Important new results on the Zak transform have been obtained analytically by Janssen, e.g., [J2; J3; J4]. The Zak transform is a unitary map of L2 (R) onto L2 ( Q), where Q = [O, 1] X [O, 1] is the unit cube in R x R. The Zak transform of 9mn has a particularly simple form, namely, Z9mn(t,w) = e2 71'i1nte271'inw Z9(t,w). It follows immediately from this formula that a Gabor system with ab = 1 is complete if and only if Z 9 =/ 0 a.e., is an orthonormal basis if and only if IZ 91 = 1 a.e., and is a frame if and only if IZ 91 is essentially constant, cf., Propsition 7.3.3. The value ab = 1 has been shown to be a critical value for Gabor systems, cf., [D1; Ri]. In particular, any Gabor system with ab> 1 must be incom- plete, and any Gabor system with ab < 1 which is a frame must be inexact. We prove in Proposition 7 .3.3 that any Gabor system with ab = 1 which is a frame must be exact, whence {9mn} is a bounded unconditional basis for L2 (R). It has been shown that if a Gabor system with ab= 1 is a frame then the mother wavelet 9 cannot be well localized both in time and frequency, 23 in particular, lltg(t)ll2 ll-r.?("Y)ll2 = oo. This is the Balian-Low theorem, cf., [Bal; Bat; BHW; DI; DJ; Low]. In this thesis we present a simple proof of a related phenomenon, namely, that if g is the mother wavelet for a Gabor frame with ab = 1 then either g is discontinuous or has poor decay at infinity, precisely, g (/:. W( C 1 0 , ? ), the Wiener amalgam space on the real line with local C0 and global ? 1 components, cf., Corollary 7.5.3. In summary, Gabor frames with ab = 1 are easily analyzed using the Zak transform, but exhibit poor localization properties. It has been shown in [DGM] (where the idea of considering Gabor or affine systems which are frames instead of orthonormal bases was introduced) that the Balian-Low phenomenon does not occur if ab < 1, i.e., Gabor systems which are inexact frames can be generated by mother wavelets which are smooth (even infinitely I " differentiable) and have good decay ( even compact support). We mention also that the Balian-Low phenomena is essentially nonexistent in a discrete setting, i.e., when considering Gabor frames for discrete signals in L2 (Z), cf., [Hl]. d. Affine systems. An affine system has the form {',Omn}m,nEZ, where and the function ',O and numbers a > 1, b > 0 are fixed. Although affine systems will not be discussed in the main part of the thesis, we include them here for completeness and comparison. A classical example is the Haar sy1tem, formed by taking ',O = X[o,1; 2] - X[1; 2 ,1], a = 2, and b = 1. The Haar system 24 forms an orthonormal basis for L2(R). In [FJ], Frazier and Jawerth introduced affine systems which are not bases, but have properties similar to frames, i.e., any element in the space can be written in terms of the affine system elements. They proved that such affine frames can be constructed in a wide range of function spaces, including the Besov and Triebel-Lizorkin spaces. Moreover, the space which the function belongs to is characterized by the behavior of the coefficients needed to write the function in terms of the affine frame elements. Later, Daubechies, Grossmann, and Meyer used Hilbert space methods to construct affine frames in L2(R), cf., [DGM]. Daubechies, Mallat, and Meyer have recently shown that it is possible to find affine systems in L2 (R) which are orthonormal bases, and which are generated by functions which are smooth and localized ( unlike the Haar system). For example, it is possible to construct a mother wavelet c.p which generates an affine orthonormal basis and which is compactly supported and k ( < oo) times differentiable, or is infinitely supported, infinitely differentiable, and exponentially decaying both in time and frequency, or is infinitely differentiable and has a compactly supported Fourier transform, cf., [D2; Mal; Mel]. Thus affine systems do not display the Balian-Low phenomenon. The existence of affine orthonormal bases has led to the introduction of fast (order N) algorithms for signal analysis, cf., [D2; Mal]. These algorithms have applications in signal processing, image processing, edge detection, etc., e.g., [Gr; KMG]. The algorithms are fast 25 and easy to implement; we have used them at The MITRE Corporation for signal analysis. e. Feichtinger and Grochenig's unified theory. A Gabor system {9mn} can be viewed as the orbit of the function g under the Schroedinger represen- tation of the Heisenberg group on a function space (see [HW) for details). An affine system { 'Pmn} can similarly be viewed as the orbit of cp under the translation/ dilation representation of the ax + b group on a function space. Thus Gabor and affine systems are structurally similar from the group repre- sentation point of view. Feichtinger and Grochenig have developed a general wavelet theory from this group representation viewpoint, e.g., [F3; F5; F6; FG2; FG3; FG4). Roughly stated, given a general representation on a gen- eral function space (satisfying certain conditions), they have shown that for a large class of mother wavelets g, any orbit {9mn} which is "dense enough" will induce representations of the functions in the function space in terms of the {9mn}- Moreover, the function space is characterized by the coefficients needed to represent functions in terms of the {9mn}? The techniques they developed to prove this general theory have also been applicable to other ar- eas, in particular, to the problem of reconstructing a band-limited signal from irregularly sampled data, e.g., [FGl]. f. Outline. In Chapter 6 we present a survey of frames ( and a dual con- cept known as sets of atoms) in Hilbert spaces, with some remarks on the extension of these concepts to Banach spaces. We discuss the representa- 26 tions of elements in the space provided by frames, and characterize when the representations will be unique, i.e, when the frame is exact. We determine the exact relationship between frames and sets of atoms, showing that while atoms are more general, in practice the two concepts will be equivalent. We prove a general stability theorem for atoms in Banach spaces, showing that the elements of a set of atoms may be perturbed by a small amount without destroying the atomic properties. In Chapter 7 we discuss Gabor systems and the Zak transform. We show that Gabor systems with ab = l can be analyzed through the use of the Zak transform. We analyze the structure of the Zak transform, and prove that it is a continuous mapping of the Wiener amalgam space W( LP, L 1 ) into the Lebesgue space LP( Q). We use this to prove a variant of the Balian-Low theorem, that a mother wavelet for a Gabor frame with ab = l cannot be continuous and have good decay at infinity, in particular, g ~ W( Co, L 1 ). We conclude by discussing some questions similar to ones which arise from the application of the Zak transform to Gabor frames. In particular, we generalize slightly a result of Boas and Pollard which shows that if finitely many elements are removed from an orthonormal basis for L2 (X) then it is always possible to find a single function to multiply the remaining elements by so that the resulting sequence is complete. We show this need not be true if infinitely many elements are deleted, and discuss some related results by other authors. 27 CHAPTER 1 NOTATION AND DEFINITIONS Section 1.1. Basic symbols. a. C is the set of complex numbers. The modulus or absolute value of z E C is denoted by Jzl, the complex conjugate by z. R is the real line thought of as the time axis, and R is its dual group, the real line as the frequency axis. Rd is d-dimensional Euclidean space, the set of d-tuples of real numbers, and Rd is its dual group. Z is the set of integers, and zd the set of d-tuples of integers. b. An element :c E Rd is written in terms of its components as :,; - ( zi, ... , :,;d). Given a, b E Rd we define All other operations on elements of Rd are to be interpreted componentwise, including logical operations. For example, if a, b E Rd then a+b - (a 1 + b1, . .. , ad + bd), ab ( a1 b1, ... , adbd), a/b (a1/b1,.,,,ad/bd), ab - ( a1 fl1 , ??? ,ad 6.i) , 28 cos a = (cosa1,,,. ,cos ad), a > b <=> ai > bi for j = 1, ... , d. An operation between a E Rd and c E R is treated by identifying c E R with ( c, ... , c) E Rd, e.g., a + c ( a1 + c, . .. , ad + c ), c/a - (c/a1, ... ,c/ad), a > c <=> ai > c for j = 1, ... , d. c. The concatenation of a E Rd, b E R k is ( a, b) = (a 1, ... , ad, b1, ... , b1;) E Rd+k. 29 Section 1.2. Special sets. a. The coordinate axes, or more precisely, the coordinate hyper- planes, in Rd are Ad= {xERd:II(x)=O}. b. The d-dimensional multiplicative group is under componentwise multiplication. The identity element of R~ is ( 1, ... , 1 ). c. The unit sphere in Rd is sd-1 - {x E Rd: lx l = 1}. d. The set of signs in Rd is { u E Rd : u i = ? 1 for j = 1, ... , d}. e. If EC Rd then E+ E+ - {x E E: x > O}. f. A rectangle in Rd is a rectangular box whose sides are parallel to the coordinate axes. Given a, b E Rd with a s; b, the open rectangle determined by a, b is d ( a, b) IT(aj,bj) {x E Rd: a< x < b}. j==l 30 We similarly define the closed rectangle [a, b] and the half-open rectangles [a , b) and ( a, bJ. The side lengths of any such rectangle are the components of the d-tuple b - a. We allow a or b to be scalars, identifying a E R with (a, ... , a) E Rd. For example, [O, b) is the rectangle with one vertex at the origin and the other at b. If both a and b are scalars then some dimensional confusion could result; however, the dimension should always be clear from context. For example, [O, 1] is a cube in Rd for any d. Given T E Rt we define 11 ,,, .,:,? [-T,T]. .._~,, , ,, ,:i P::'I,. .;,. JI 31 Section 1.3. Functions. = { if XE E, a. The characteristic function of a set Eis XE(x) l, o, if X ? E. 1, if X = y, The Kronecker delta is Dzy = { 0, if X =F y. b. A real-valued function f on a set Eis positive if f(t) > 0 fort EE. It is nonnegative if f(t) 2: 0 fort EE. c. A function f:Rd---+ C is P-periodic, where PE Rd, if f(t+P) = J(t) d. A function f:Rd---+ C is symmetric if f(t) = f(-t) fort E Rd, radial if J(s) = f(t) whenever isl = !ti, and even if J(ut) = f(t) fort E Rd and ~I u E nd. ~: These three notions are equivalent if d = 1 but not if d > 1. Every 'I' radial function is even, and every even function is symmetric. If d > 1 then I ? JI the function f (t ) = II I( t) I is even but not radial, and f ( t) = sign( t 1 ) ? sign( t2) Ji! -? is symmetric but not even. e. A function f: Rd ---+ C is rectangular if there exist functions Ji: R---+ C such that d J(t) II 1;(t1) j=l f. A real-valued function f is (rectangularly) decreasing on a set EC Rd if givens, t E E, s < t => f(s) 2: J(t). In other words, f is decreasing in each component. f is strictly decreasing if f(s) > f(t) when s < t. We similarly define increasing and strictly 32 ,, increasing. g. Given a real-valued function f on Ri, its least decreasing majorant f* is J*(t) = sup f(s). s>t Its greatest decreasing minorant f* is Clearly f* ~ f ::; J*, and f is decreasing if and only if f = f* = f*? If f is rectangular then f*(t) = IT~ f/(t;) and f*(t) = rr: J;*(t;). h. The following function spaces are defined specifically for functions on Rd; other function spaces are defined in Section 1.7. Given k E zd with k ~ 0 I I we define J.. C(Rd) {f : f is continuous}, Cc(Rd) - {f E C(Rd) : supp(!) is compact}, where C00 (Rd) and C~(Rd) are defined analogously. The Schwartz space of rapidly decreasing functions is S(Rd) = {! E C 00(Rd): sup IIT(tk)8af(t)1 < 00 for k,a E zd,k,a ~ o}. tER4 33 The space of tempered distributions, denoted S'(Rd), is the topological dual of S. 34 Section 1.4. Convergence. Given a normed linear space X and a sequence {xn}nez+ of elements of X, we say that the series E Xn converges to x E X, and write E Xn = x, if SN -. x, where SN = E:=I Xn- The series converges unconditionally if E Xt3(n) converges for every permutation /3 of Z+. It converges absolutely if E llxn/1 < oo. Absolute convergence implies unconditional convergence. If X is finite-dimensional, the converse is also true. LEMMA 1.4.1. Given a no1?med linear space X, the following statements are equivalent. .,, ..1 ,. a. X is complete. b. If {xn}nEZ+ C X and :E llxnll < oo then E Xn converges in X. LEMMA 1.4.2 [S]. Given a sequence {xn}nEZ+ m a Banach space X, the following statements are equivalent. a. E Xn converges unconditionally. b. x = limF EneF Xn exists, where the limit is with respect to the net of finite subsets of Z+ ordered by inclusion. In other words, for every e > 0 there is a finite set G C Z+ such that llx - EneF xnll < e for every finite F C Z+ with F :::> G. c. For each e > 0 there is an N E Z+ such that for each finite F C Z+ with min(F) > N we have II EneF xnll < e, d. E Xni converges fo1? every increasing sequence O < n1 < n 2 < .... 35 e. ~ O"n:Z:n converges for every choice of signs O"n = ?1. f. E CnXn converges for every bounded sequence of scalars {en}, In ca.se these hold, E Xf3(n) = E Xn for every permutation /3 of Z+. LEMMA 1.4.3. Given a Banach space X a.nd a sequence {xn}nEZ+ C X . a. If x = E Xn converges then !!xii $ E llxnll $ oo. b. If E llxnll < oo then x = E Xn converges unconditionally. PROOF: a. Given e > O, there exists by definition an N > 0 such that !Ix - E~ xnll $ e. Therefore, N N N CXl llxll $ llx - L :Z:nll + IIL :Z:nll < e + L llxnll < e + L 11:z:nll? 1 1 1 1 Letting e -+ 0 gives the result. b. Follows immediately from the triangle inequality and Lemma 1.4.2. I 36 Section 1.5. Operators. a. Assume X and Y are Banach spaces, and that S: X--+ Y. Sis linear if S(ax +by)= aSx + bSy for x, y EX and a, b EC. S is injective if Sx =f. Sy whenever x =f. y. The range of Sis Range(S) = {Sx: x EX}. S is surjective if Range( S) = Y. i S is bijective if it is both injective and surjective. , J ,[ .,,? The norm of S is IJSII = sup {1/SxllY : x EX, llxllx = I}. r' f S ' is bounded if I/SIi < oo. A linear operator is bounded if and only if it is continuous, i.e., if Xn --+ x implies Sxn --+ Sx. ,, ,. .. The adjoint of S is the unique operator S': Y' --+ X' such that (Sx, y') = ,:.i ,,,,:,: (x,S'y') for all x EX and y' E Y', where X', Y' are the Banach space duals :.JI of X, Y, respectively. S is invertible, or a topological isomorphism, if S is linear, bijective, continuous, and s-1 : Y --+ X is continuous. Sis an isometry if //Sxlly = /Ix/Ix for all x EX. S is unitary if it is a linear bijective isometry. L(X, Y) = {S:X--+ Y: Sis linear and continuous}. L(X) = L(X,X). b. Assume H is a Hilbert space and S, T: H ---+ H. S is self-adjoint if (Sx, y) = (x, Sy) for x, y E H. S is positive, denoted S ~ O, if (Sx, x) > 0 for x E H. All positive 37 operators are self-adjoint. S ~ T if S - T ~ o. c. For functions f on Rd we define the following operators. Translation: Taf(t) - f(t - a), for a E Rd, Modulation: Eaf(t) - e2wia?t f(t), for a E Rd, Dilation: Daf(t) - f(t/a), for a ER~. We also use the symbol Ea to refer to the exponential function Ea(t) = e2wia?t, where a, t E Rd. 38 Section 1.6. Topological groups. Although some sections of this thesis are written in terms of abstract topo- logical groups, in practice we use only the additive and multiplicative groups on Rd. a. The set Rd is a topological group under componentwise addition, with Haar measure equaling Lebesgue measure dt. The set Rd will always be assumed to have this operation and measure. The group translation operator is ordinary translation: T0 f(t) = f(t - a). The measure of a set EC Rd with respect to Lebesgue measure is denoted by IEI, The sets R~ and Ri are topological groups under componentwise multi- plication, with Haar measure dt/lII(t)I, The sets R~ and Ri will always be assumed to have this operation and measure. The group translation operator for these groups is dilation: D0 f(t) = f(t/a). The measure of a set EC R~ with respect to this Haar measure is denoted by IEI, Integrals with unspecified limits are assumed to be over Rd with respect to Lebesgue measure dt. b. We point out the following facts about the multiplicative group R~. Compact sets in R~ are bounded away from both oo and the coordinate axes. A connected compact set is entirely contained in one quadrant of R~. Haar measure dt/lII(t)I is dilation invariant. Given EC R~, IEI .. = 0 if and only if IEI = 0. Therefore the term almo8t everywhere (a.e.) has the same meaning in the additive and multiplicative 39 groups. To see this, assume EC R: with jEI = 0 is compact and contained in one quadrant of Rd, say Ri. Then E C [a , b] C Ri, so f dt 1 / jEj. = JE IIT(t)l ~ IIT(a)I JE dt = O. The general case follows since R: is u-finite, and the converse is similar. ,. . ft ?' :i ,I.' ,. ,. 11 40 Section 1. 7. Function spaces. Let G be a u-finite, locally compact group with left Haar measure dx. A positive function w on G, i.e., w: G ---+ R+, is a weight on G. In this thesis, all functions defined on topological groups or measure spaces are assumed to be measurable. a. Given 1 < p < oo and a weight w on G, we define the weighted LP-space ,. J"I Lt(G) - {/: G---+ C: llfl!L~(G) < oo}, d ',"., !1 :l where .I? ?' (11/(x)IP w(x) dx) I/p, .. if 1 $ p < oo, .. 1/IIL~(G) = a { :i ess sup 1/(x)I w(x), if p = oo. ,,,,,. :i:EG ,, p If w = 1 then we write LP ( G) = Li ( G). When G is understood we write Lt or LP. We let II ? IIP = II ? JILP. When G is countable and dx is counting measure we write ft( G) instead of Lt( G). Lt( G) is a Banach space for 1 $ p $ oo. The dual index to p is p' = p/(p -1), i.e.,?+;, = 1. The dual weight tow is w' = w 1-P'. We have I (Lt)'= L~, for 1 ~ p < oo, where the prime denotes the Banach space dual and the duality is defined by (f,g) = Lf(x)g(x)dx for f E Lt(G ) , g E Lwp', (G). Note that L 2 (G) is a Hilbert space under this inner product. 41 b. We define the following additional spaces of functions on G. Lfoc(G) - {i: G --t C: i ? XK E LP(G), all compact KC G}, C(G) - {i: G --t C : i is continuous}, Cc(G) - {i E C( G) : supp(!) is compact}, Cb(G) {i E C( G) : i is bounded}, Co(G) - {i E C( G) : i vanishes at infinity}, where vanishing at infinity means that for each E: > 0 there exists a compact KC G such that li(x)I < E: for all x (J. K. (Cb(G), 11 ? lloo) and (Co(G), II? lloo) are Banach spaces; Cc(G) is dense in (LP( G), II ? lip) for 1 $ p < oo, and in ( Co( G), II ? lloo), c. A Banach function space, or BF-space, on G is a Banach space B continuously embedded into Lloc(G), i.e., for each compact KC G there is a CK > 0 such that Iii? XKIIL1(G) S CK lli!IB for each i E B. A BF-space B is solid if given i, g E B with Iii s; IYI a.e. we have llillB $ IIYIIB? The spaces Lt(G) and Co(G) are solid. Lt(G) possesses the stronger property that if i E Lf J G) and g E B = L~( G) with Iii $ Jgl 0 a.e. then i E B and lli llB $ IIYIIB- Co(G) need not satisfy this, e.g., take G=Rd. d. Given a E G, the left and right group translation operators are and e. Let B be a Banach function space on G. 42 B is closed under left translations if La(B) CB for each a E G. B is left translation invariant if it is closed under left translations and La: B --+ B is continuous for each a E G. If each La is an isometry then B is left translation isometric. Translation is strongly continuous in B if lima--+b \\Laf - Lbfl\B = 0 for all f E Band b E G, where the limit is taken in the group topology sense, i.e., for each e; > 0 there is a neighborhood U of b such that \\Laf - Lb/1\B < e: for a E U, cf., Section 1.9a. B is left homogeneous if it is left translation isometric and translation is strongly continuous in B. B is a left Segal algebra if it is left homogeneous and is dense in L 1 ( G) in the L 1-norm. Similar definitions are made with right in place of left. If the term left or right is omitted, it is assumed that both hold, for example, if G is abelian. f. The following inclusions hold for fP. If O < p ~ q ~ oo then fP C fq, with II ? \\tP ~ II ? \\tq ? For O < p < 1, fP is not a Banach space, but is a complete metric space with distance defined by d(f,g) = \If - gl\~. The triangle inequality for this distance is equivalent to the estimate g. If EC G has finite measure and 1 ~ p ~ q < oo then 43 This also holds for 1 :S p < q = oo if the right-hand side is replaced by ess suptEE lf(t)I- Equivalently, for all 1 :Sp '.Sq '.S oo, with the interpretation 1/oo = 0. 44 Section 1.8. The Fourier transform. a. The Fourier transform of a function f E L1 (Rd) is }(-r) = Jf (t) e-21ri-,?t dt, defined for "Y E Rd. The inverse Fourier transform is ](-r) = f(--y) = jJ(t)e21ri-,-tdt. The Fourier transform of J E L2 (Rd) is J = limn-oo (J ? XR,. )J\, where the limit is in the L2-norm. b. The Plancherel formula is IIJIIL2(a.t) = IVIIL2(:fl..t) = II/IIL2(:fl..t)? The Parseval formula is (f,g) = (f,g) = (/,g), where(?,?) is the L2(Rd) inner product. The inversion formula is f = JJ\V = rJ\ for f E L2 (Rd). If f E S(Rd) then we have the Poisson summation formula L f(k) - L f(k). kEZ.t kEZ.t c. We have the formulas and 45 Section 1.9. Group representations. Let G be a locally compact group and X a Banach space. a. A representation of G on Xis a homomorphism of G into L(X), i.e., a mapping U: G--+ L(X) such that for x,y E G. U is unitary if each Uz: X --+ X is a unitary operator. U is strongly continuous if limz-+y Uz = Uy, where the limit is taken in the strong operator topology. That is, for all / EX, where this limit is in the group topology. b. If X = H, a Hilbert space, then we make the following additional definitions. A element g EH is admissible if fa l(Uzg,g}l 2 dx < oo. g is cyclic if span{Uzg }zeG is dense in H. U is square-integrable if there exists an admissible g E H\{0}. U is irreducible if every g E H\{O} is cyclic. c. The following result is well-known, e.g., [GMP]. PROPOSITION 1.9.1. If U is a square-integrable and irreducible representa- tion of a locally compact group G on a Hilbert space H then there exists a 46 unique self-adjoint positive operator C: Domain(C) ~ H such that a. Domain{ C) = {9 E H : 9 is admissible}, b. given any / 1, /2 E H and any admissible 91, 92 E H, Moreover, if G is unimodular then C is a multiple of the identity. Setting Ji = /2 = 91 = 92 = 9 in Proposition 1.9.1, we obtain Setting Ji = /2 = f and 91 = 92 = 9, we obtain 47 ? I .~ ' .. PART II ..I'!.: . WIENER AMALGAM SPACES ~' ? 48 CHAPTER 2 WIENER AMALGAM SPACES In this chapter we discuss the far-reaching generalization of amalgam spaces derived by Feichtinger, e.g., [F2; F8]. Given Banach function spaces B, C on a locally compact group G, satisfying certain conditions, he defined spaces W(B, C) of distributions which are, roughly speaking, locally in Bin globally in 0. The space W(LP(R),Lq(R)) coincides with the standard amalgam space defined in (0.1.1). While each W(B, C) can be described in terms of a discrete-type norm like (0.1.1), the fundamental norm describing the local/global properties is a continuous-type norm (cf., Sections 2.2 and 2.4). These equivalent continuous and discrete norms provide flexibility in using the W(B, C) in applications. Feichtinger calls the spaces W(B, C) Wiener-type spaces; following a sug- gestion of J. Benedetto, and in order to promote the link between Feichtinger's generalization and amalgams occuring previously in the literature, we call them Wiener amalgam spaces. Wiener amalgam spaces lie at the heart of many of the main results of this thesis, especially those in Part III (Generalized Harmonic Analysis). In those chapters, we use the Wiener amalgam spaces W(LP(R:), Lq(R:)), on the multiplicative d-dimensional group R:. It is the discrete-type norms on this space which provide the machinery for our major results. Amalgams play 49 a smaller, but still important role in Part IV (Wavelet Theory). There we use the standard amalgam spaces W(LP(Rd),Lq(Rd)) on the additive group Rd. The purpose of this chapter is to review fundamental facts about the Wiener amalgams W(B, C). As noted above, the main results in this thesis use only the cases B = LP(G), C ?= Lq(G); some minor results use B = L~(G) or C = 0 0 ( G). We therefore present the Feichtinger theory only for the spaces W(L!(G),Li(G)). This results in a considerable technical simplification of the general W(B, C) theory. This chapter can therefore be regarded as an elementary introduction to the general theory presented in [F8]. The results in this chapter are known; we have collected results and proofs from many sources, including [Fl-F8; FG; Ho; FS; Wa] and others. The credit for this chapter therefore belongs primarily to Feichtinger and secon- darily to others; we have synthesized their results into a single expository chapter. We now outline this chapter by sections. In Section 2.1 we characterize those weights w for which the weighted LP space Lt( G) is translation invariant. Section 2.2 contains the basic definitions of the Wiener amalgam spaces in terms of continuous-type norms, and proofs of fundamental properties such as completeness and translation invariance. In Section 2.3 we determine various inclusion relations between the spaces W(L~(G), Li(G)). 50 In Section 2.4 we derive equivalent discrete norms for Wiener amalgam spaces. These norms are the ones which will be used in the proofs of the major results in Parts III and IV. Finally, in Section 2.5 we prove duality relationships between the amalgam spaces. We assume throughout this chapter that G is a u-fi.nite, locally compact group. Since in later chapters we use only G = Rd or G = R~, we assume for simplicity that G is unimodular, i.e., left and right Haar measure coincide. We denote this Haar measure by dz, the identity element bye, the left group translation operator by Laf(z) = /(a-1 z), and the right group translation operator by Raf(z) = /(za-1 ), cf., Section 1.7. The measure of a set ECG with respect to Haar measure is denoted by !El. A positive function w: G-+ R+ is called a weight. 51 , J , I I Section 2.1. Moderate weights. In this section we characterize the class of weights w for which the Banach function space Lt( G) is translation invariant. The results in this section are known. In particular, the proofs given here are extensions to locally compact groups of Walnut's proofs on the additive group Rd (Wa]. See also the original results in [Ed; Gau]. DEFINITION 2 .1.1. a. A weight m: G ---+ R+ is submultiplicative if m( e) = 1 and m(xy) ~ m(x)m(y) for x, y E G. r b. A weight w: G---+ R+ is right moderate if there exists a submultiplica- tive function m such that w(xy) ~ w(x)m(y) for x, y E G. ,, Corresponding definitions and theorems for left moderate weights are as- ,,,, ,: .: sumed throughout this section. If the term left or right is omitted, it is assumed that both hold. PROPOSITION 2.1.2. If w is right moderate with associated submultiplicative function m then w(e)/m(x- 1 ) ~ w(x) ~ w(e)m(x) for all x E G. PROOF: We compute w(x) = w(ex) ~ w(e)m(x) and w(e) = w(xx-1 ) < PROPOSITION 2.1.3. Given wright moderate with associated submultiplica- function m"". b. If r < 0 then wr is right moderate with associated submultiplicative function mr, where m(x) = m(:z:-1 ). PROOF: Part a is clear, and therefore for part b we need only consider r = -1. That m is submultiplicative is also clear, and that w-1 is right moderate follows immediately from the computation THEOREM 2 .1.4. Submultiplicative functions are locally bounded. PROOF: Assume mis submultiplicative. We claim first that if mis bounded on any open neighborhood of the identity then it is bounded on every compact set. To see this, assume mis bounded on some open U containing e, and let K be a compact set. Then K C LJf x1cU for some z1, ... , :z:N E G. Let R = max{m(:z:k)}, If :z: E K then :z: = :z:,ey for some k and some y E U, so m(:z:) = m(:z:1cy) ~ m(xk)m(y) ~ Rllm ? Xulloo? Therefore mis bounded on K, as claimed. Now suppose that m was unbounded on every open neighborhood of e. Let U be an open neighborhood of e with compact closure, such that U = u-1 ? Since laU ~ UI = LIX au - Xul = IILaXu - Xullu(G) and left translation is strongly continuous in L 1 ( G), there exists a neighbor- hood V of e such that (2.1.1) laU ~UI < ?IUI 53 for a EV. Now, for each N E Z+ there exists by assumption an z N E V such that m(xN) ~ N 2. Therefore, given x E G we have so either m(:z:Nx-1 ) >Nor m(x) ~ N. Defining AN = {x EU: m(x) ~ N}, we therefore have (2.1.2) since if y E U and 1 y = XNX- for some x E U\AN then m(x) < N, so m(y) = m(:z:Nz-1) ~ N, whence y E AN. Since X\Y:) X\Z => X n Z ::, X\Y and it follows that (2.1.3) xN (U\AN )- 1 nu :) XN(U\ANr 1 \(xNu-1 \U) XN (U\AN)-1 \ (xNU\U) :) ZN (U\AN )-1 \ (xNU AU). 54 Since XN E V we therefore have from (2.1.1), (2.1.2), and (2.1.3) that IANI > lxN (U\AN )-1 n UI > lxN (U\AN )-1 \ (xNU L\. U)I > lxN(U\AN)-1 1 - lxNU L\.UI IU\ANI - lxNU L\.UI > IUI - IANI - ?IUI, ~ whence ,lP, :: IANI ~ ?IUI. Since the sets AN are nested in U, this implies In ANI ~ ?IUI > O. However ' m is finite-valued, so nAN = 0, a contradiction. I ,, ,I,? ,, ,: COROLLARY 2.1.5. Every right moderate function is locally bounded. ,: PROOF: Assume w was right moderate but unbounded on some compact set K, and fix any :z: E G. Let m be the submultiplicative function associated with w. By Theorem 2.1.4, mis locally bounded, so M = llm?Xz-1Klloo < 00 ? Now, given R > 0 there exists y EK such that w(y) > RM. Therefore, RM < w(y) $ w(x)m(x-1 y) $ w(x)M. As x and R are arbitrary, this is a contradiction. I THEOREM 2.1.6. Given a positive w E L:0 c(G), the following statements are equivalent. a. w is right moderate. 55 b. Li(G) is closed under right translations for some (and therefore every) 1 ::; p:::; 00. c. Li(G) is right translation invariant for some (and therefore every) 1 ~ p:::; 00. d. For each compact K C G, w(:z:y) A(K) sup ( ) < oo. zEG,yEK W Z e. Given any compact set KC G there exists a constant B = B(K) such that sup w < B inf w zK zK for every :z: E G. f. Given any compact set K C G there exist constants C = C(K), D = D(K) such that Cw(y) < ( w(t)dt < Dw(y) fzK for al.l y E :z:K. g. Given any compact set KC G and given k EK there exist constants E = E(K,k), F = F(K,k) such that Ew(xk) < ( w(t)dt < Fw(xk) JzK for al.l :z: E G. 56 PROOF: a=> c. Assume w is right moderate with associated sub ult" Ii . Ill: 1p cative function m. Given 1 < p < oo, f E Yw, and a E G, we then have Llf (z)IPw(:z:a)dz < m(a) Llf (z)IPw(x) dz < m(a) llfllf,. ? "' Thus Ra maps L~ into itself, and does so continuously, with IIRall < m(a)l/p_ The case p = oo is similar, with the result IIRall $ m(a). c => a. Assume that c holds, and fix 1 $ P < oo. For a E G define m(a) = IIRallP, Note that m(e) = IIIIIP = 1 and so m is submultiplicative. We show now that w is right moderate with mas associated submultiplica- tive function. Fix any a E G and f E Yu,. Then $ m(a) II/lift = m(a) L1 /(:z:)IP w(x) d:z:. 57 Since this is true for every I E Lt, we have w( :z:a) $ w( :z:) m( a) for a.e. :z: E G , sow is moderate. The case p = oo is similar (set m(a) = IIRal!). b => c. Assume ~ is closed under right translations for some 1 $ p $ ? 00 Given a E G, assume In E Yw are such that In__. IE~ and Rain__. E y 9 1D as n __. oo. Then we can find a subsequence {/n?} where both convergences are pointwise a.e. Then Rain,. __. Raf, 9 pointwise a.e., whence Ra/= g a.e. Ra is therefore continuous by the closed graph theorem. a => d. Assume w is right moderate with associated submultiplicative function m. By Theorem 2.1.4 we have m E L:c, so if y EK, a compact set in G, and :z: E G then w(:z:y) $ w(x)m(y) < w(:z:) !Im? XKll(X)? Thus A(K) $ llm ? XK\10() < oo. d => c. Assumed holds, and let K C G be compact. Given a E K and I E Lt( G), where 1 < p < oo, we have L1 1(:z:)IP ~~i w(:z:)d:z: $_ A(K) L1 1(:z:)jPw(:z:)dz - A(K) lllllti ? Therefore Ra maps ~ into itself, and does so continuously, with IIRall < A(K) 1IP. The case p = oo is similar. 58 d => e. Assume d holds, and let K C G be a compact set and x any element of G. Set L = KU x-1 , and note that L is both compact and symmetric (i.e., L-1 = L). If y E xK then x- 1 y EK CL, so y-1 x EL. Hence, 1 ( ) w(xx- y) ( ) ( ) w Y - w(x) w x < A(L)w x and ( w(yy-l X) ( ) ( ) ( w x ) w(y) w y $ A L w y), so e => d. Assume e holds, and let K C G be compact. Let L :) K be compact, with e E L. Given x E Gandy EK we then have w(xy) < sup w < sup w < B(L) inf w < B(L)w(xe) B(L)w(x) zK zL zL since e E L. Therefore, A(K) $ B(L) < oo. e ? f. Assume e holds, let K C G be compact, and let x be any element of G. Then 1f1) suKp w ~ lxKI inf w $ 1 w $ lxKI sup w $ IKI B(K) inf w. 2 zK zK zK zK g => e. Assume g holds, and let K C G be compact and x any element of G. Then L = K U x-1 u {e } , L' = LL, and L" = L' L' are all compact symmetric sets containing e. The symmetry implies that xL C yL' C xL" 59 and yL C xL' C yL" for y E xL. Therefore, < F(L",e) F(L" e) E(L':e) E(L',e) F(L",e) 1 f < E(L',e) w(x)}zL'w, w(x) FE((LL",,' ee))so ~ w(y) for y E xL. Similarly, > _1 r w - w(y) }yL > E(L, e) F(L' ) F(L',e) ,e > E(L, e) 1 f F(L',e) w(x) JzL' w. Thu s w ( x ) ~ E(L,e) ( ) .c L F(L' ,e) w y ior y E x , so sup w < sup w zK zL < F(L',e) w(x) E(L,e) < -F-(L-'-, -e), F--(L-"-, ,e-) . f Ill W E(L,e)E(L',e) zL < -F(-L'", e-) F-(-L"-, e-) . mf w. I E(L,e)E(L',e) zK PROPOSITION 2.1. 7. Every right moderate weight is equivalent to a continu- ous right moderate weigl1t in the sense that if w is right moderate then there exists a continuous right moderate v and constants A, B > 0 such that Av(x) ~ w(x) ~ Bv(x) 60 for all z E G. PROOF: Assume w is right moderate, and let k E Oc(G) be any function such that k > 0 and JG k = 1. Let K ::) supp( k) be a compact symmetric neighborhood of e. Since w is locally bounded, we can define Clearly v is positive and continuous, and v(xy) = la w(t:cy)k(r1 )dt =::; m(y) la w(t:c)k(t-1 )dt = v(x)m(y), so v is right moderate. Also, v(x) - Lw (t:i:) k(r 1 ) dt < sup w ? l k( t-1 ) dt zK K < B(K) inf w zK < B(K)w(:c), where B(K) is as in Theorem 2.1.6e. Similarly, v(x) ~ B(Kt1 w(x), so we are done. I 61 Section 2.2. Definition and basic properties. In this section we define and derive basic properties of the spaces W ( B, C). Our proofs will hold when B, C are weighted ?P spaces L{:,(G), where 1 ::::; p ::::; oo and w: G -----t R+? For these spaces, integrability (local and global) is the only defining factor. This simplifies the proofs from the general abstract case; we attempt to indicate what technical modifications are necessary to cover the general case. Note that ?{:,( G) is solid in the sense of Section 1. 7 c, and is right translation invariant if and only if w is right moderate (Theorem 2.1.6c). The primary space we are interested in other than the weighted ?P spaces is C0 ( G), the continuous functions on G vanishing at infinity. With the weighted LP spaces as a model, we define B1oc {f: G -----t C: f ? XK EB for every compact KC G}. REMARK 2.2.1. This definition is not the proper one to make if B has prop- erties other than integrability, e.g., smoothness. For B = C0 (G) it would be appropriate to take B10 c {f E M(G) = Cc(G)': fcp EB for every cp E Cc(G)}, with corresponding technical difficulties added to the proofs. For the general case we would assume that there is a homogeneous Banach space A such that: a. A is continuously embedded into (Cb(G), II? lloo). b. A is a regular Banach algebra under pointwise multiplication. c. A is closed under complex conjugation. 62 d. B is continuously contained in Ac', where Ac = {f E A : supp(!) is compact}. e. A is a Banach module over B with respect to pointwise multiplication, i.e., if f E A and g E B then Jg EB with 1/fgl/B :$ llfllA ll91/B? Then we would define B1oc = {! E Ac': fcp EB for cp E Ac}, This can be shown to be independent of the choice of A. DEFINITION 2.2.2. Fix a compact set Q C G with nonempty interior. For f ; ' E B1oc and X E G define .I ,? The Wiener amalgam space W(B,C) is W(B, C) = {! E B1oc: F1 EC}, with norm We refer to B as the local component and C as the global con1ponent of W(B,C). REMARK 2.2 .3 . For the general case, we would define F1(x) = inf {1/gl/B: g EB and gcp = fcp for cp E Ac with supp(cp) C xQ}, and again set /lf/lw(B,C) = I/F1//c. EXAMPLE 2.2.4. We compare W(L 00 ,L1 ) to W(C ,L10 ) . 63 00 Since both L and Co are equipped with the L=-norm, W(L 00 , L1 ) and 1 W(Co,L ) have the same norm. However, the definitions of (L00 )1oc and (Co)loc differ, so they are distinct spaces. In fact, 1 W(Co,L ) = {/ E W(L00 ,L1 ): f is continuous}. THEOREM 2.2.5. a. W(B, C) is a Banach space. b. If C is solid and right translation invariant then W(B, C) is independent of the choice of Q, i.e., different choices of Q define the same space with equivalent norms. PROOF: a. That II? \\w(B,C) is a norm is clear, so we prove that W(B,C) is complete in this norm. Assume {fn}neZ+ C W(B, C) with}:: llfnllW(B,C) < ., ",,.". , oo. By Lemma 1.4.1 it suffices to prove that }:: fn converges to an element .,:.:.: of W(B,C). Now,}:: llfn\\w(B,O) =}:: \\Ftn\lo and C is complete, so }::F/n must converge to an element of C. Therefore, for a.e. :c E G. Since B is also complete, 1: fn ? Xzq must converge to an element 9z E B. Clearly 9:i: = g11 a.e. on xQ n yQ, so we can define a function g a.e. by g(t) = 9:i:(t) fort E xQ, i.e., g ? Xzq = 9z? Applying Lemma 1.4.3 twice, we have \\g\\w(B,O) - 111\g ? Xzq IIB lie 64 < LIi llfn ? XzQIIB lie - L llfnllW(B,C) < oo, so g E W(B, C). A similar computation shows N llo - ~ fnllw(B,c)-+ O as N-+ oo, so Lin converges in W(B,C) tog. Therefore W(B,O) is complete. b. Assume that O is solid and right translation invariant, and let Q , q be 1 2 two compact subsets of G with nonempty interiors. Then we can find points :z: 1 , ??? , :z: N E G such that ,.., . ?,,:,?: "' For :z: E G we therefore have Ff2(:z:) - II/? XzQ2 lls < Iii? Xuzzr.Q1 lls N < IIJ . ~ Xzzr.Qi 11B N < L llf ?Xzzr.Q1IIB l N - L F?l (:Z ::Z:Js) l N - L (Rz;1Ff1) (:z:). 1 65 Since both FJ 1 and :E R'J)-;;1FJ1 are elements of C and C is solid, we have N IIFJ2 llo $ II~ Rz-;;1FJ1 Ila 1 N < ~ 1 IIRz; 1 FJ lie 1 N $ ~ M 1 IIFJ lie 1 - MN 1 IIFJ lie, where M = max{IIRz;1ll0} < oo by the translation invariance of C. A symmetric argument gives the reverse inequality and completes the proof. I REMARK 2.2.6. General amalgams W(B, C) can be shown to be also inde- pendent of the choice of Banach algebra A ( cf., Remark 2.2.1 ). We assume from now onwards that C is solid and right translation invariant. LEMMA 2.2.7. LaXE = XaE, RaXE = XEa? PROOF: The second statement is similar. I PROPOSITION 2.2.8. If B and C are left translation invariant then so is W(B, C), with If B, Care left translation isometric then so is W(B,C). 66 PROOF: Assume f E W(B, C). As B is left translation invari .. T'Ot h -u we ave La.f EB, so FLa.l EC. Now, (2.2.1) - IILa(f ? Xa.-1.q)lls < IILallB llf ? Xa.-iaqllB = IILcallB Fi(a-1 :z:) Since C is left translation invariant, LaFJ E C. Therefore, since C is solid ' IILcafllw(B,C) = IIFLa.Jllc $ II LallB IILaF1llc The translation isometric case is similar. I PROPOSITION 2.2.9. Ifleft translation is strongly continuous in B, C is trans- lation invariant, and Cc( G) is dense in C, then left translation is strongly continuous in W(B, C). PROOF: Fix f E W(B,C) and e > 0. Then there exists a k E Cc(G) such that IIF1 ? (1 - k)llc < e. Let K = supp(k). For a E G We then have (2.2.2) IILcaFJ ? (1 - k)llc < IILa(F1 ? (1 - k))llc + ll(LaFJ) ? (1 - k) - La(FJ ? (1- k))llo 67 < IILallc IIF1 ? (1 - k)llc + IILallc IIF1 ? La-1(1- k) - Fi? (1- k)llc ::; ? IILallc + IILallc IIF1 ? (La-ik - k)llc ::; ? IILa lie + II La lie IIF1llc l!La-1k - kll 00 , where the last inequality follows from the fact that C is solid and both F I and Fi . (L11 -1k - k) E C. Since IILa-ik - klloo ----1- 0 as a ----1- e, we can find a neighborhood U of e ( with compact closure) such that (2.2.3) for all a E U. Now, IILallB and IILallc are locally bounded as functions of a since they are submultiplicative functions on G (Theorem 2.1.4). Therefore, M = supaEU IILallB, IILallc < oo, which, combined with (2.2.2) and (2.2.3), gives (2.2.4) for a E U. Combining (2.2.4) with (2.2.1), we have IIFL .. J ? (1 - k)llc ::; IILallB IILaFJ ? (1 - k)llc M 2 ::; ? (1 + IIF1ll0) for a E U. Since FL.J-J ~ FLa.l + Fi and supp(k) = K, we therefore have for a EU that 68 ! ? IILaf - fllw(B,C) < IIFL,.f-1 ? (1 - k)llc + IIFL.f-1 ? kilo < IIFL,.f ? (1 - k)llc + IIF1 ? (1 - k)llc + llkllc sup FL,.f-1(x) zEK < Re + e + llkllc sup ll(Laf - f) ? XzqllB? zEK The result now follows from the fact that K is compact and left translation is strongly continuous in B. I COROLLARY 2.2.10. If Bis left homogeneous, C is translation isometric, and Cc(G) is dense in C, then W(B,C) is left homogeneous. ,?.' EXAMPLE 2.2.11. W(B,LP(G)) is left homogeneous for 1::; p < oo. ,, ,,,, .:;. 69 Section 2.3. Inclusion relations. In this section we derive some inclusion relations between the amalgams W(L~, Li). We assume throughout the remainder ofthis chapter that v, ware weights on G with wright moderate (so, in particular, Li is right translation invariant by Theorem 2.1.6). Note that This expression is sometimes used as the defining norm for Lt, rather than L!:,,,, as we use it. Some of the results in this section would be easier to state if we adopted this alternate definition of~, but it will be convenient in the main part of the thesis to keep the w's on the "outside". Recall from Proposition 2.1.3 that wP is right moderate if and only if w is. For simplicity and consistency in dealing with the case p = oo we let w 00 = A moderate weight can be transferred between the local and global com- ponents, as follows. PROPOSITION 2.3.1. Given 1 Sp, q S oo, PROOF: Assume 1 S p, q < oo. Since w is right moderate, there exists by Theorem 2.1.6e a constant B such that B-1w(t) S w(a::) S Bw(t) for all 70 x E G a.nd t E zQ. Therefore, ll/llw(L1'p,L9 9 ) 1111/ ?XzqllLPP IIL" 11 W II w9 - (L (t IJ(t)v(t)I' dt) 11q/p w(~)? d~) ' < (L (/.Q If( t ) v( t) Bw (t) 1? dtr ? u r? The opposite inequality, and the rema.ining cases, are similar. I When the local and global components are comparable, we have the follow- mg. PROPOSITION 2.3.2. Given 1 ~ p ~ co, W(~, L~) = L~10 , with equivalence of norms. If e E Q and IQI = 1 then II? llw(L!,LP) - 11 ? IILt? PROOF: Without loss of generality assume e E Q. For 1 < p < co we have 11/ll~(LP LP) = v, '" Jfa Jfz q lf(t)JPv(t)dt w(x)dx = LLIJ(t)jPTJ(t)Xzq(t)w(z)dtdx - L 1/(t)jP v(t) L w(x) X,q-1(:i:) dx dt. 71 Since w is right moderate, there exist by Theorem 2.1.6? constants C, D > 0 such that Cw(t) $ f w(x)dx $ Dw(t) ltq- 1 fort E G (note that if w = 1 then C = D = IQ-1 I = !QI). Thus, llfll~(L! ,L!,) $ D l lf(t)IP v(t) w(t) dt = D llfllf:,.,. The opposite inequality, and the case p = oo, are similar. I One simple inclusion relation is the following. PROPOSITION 2.3.3. Assume Bis solid. If 1 $ p < oo and w E L 1(G), or jf p = oo and w E L 00(G), then W(B,~) :J B. PROOF: Assume 1 $ p < oo and w E L1 ? If f EB then (L II!? x.qll';, w(x)d,:) ?I? < (L II/II':. w( x) dx )'1' llfllB llwl!L 1 ? The remaining case is similar. I The standard inclusion relations for LP spaces on compact sets imply in- clusion relations with respect to the local components of Wiener spaces. PROPOSITION 2.3.4. Given 1 $ p $ q $ oo, W(L!P, C) :) W(L!11 l C) 72 r ' J J with 1 1 II? llw(LJJp,C) ~ !QI,-, II? llw(L' C)? V ~? PROOF: From Section 1.7g, The result now follows from the solidity of C. I ;. The following lemma is an integral version of Minkowski's inequality, e.g., .I.' [WZ, p. 143]. ,,, ,? I . LEMMA 2.3.5. Given measure spaces (X,?) and(Y,11), and given 1 ~ p < oo. ' If Fis measurable and nonnegative on Xx Y then ([ (L F( z, y) d?.( z >)' dv(y) )"' 5 L( [ F( z, y 'f dv(y >)"' d?.( z ). PROPOSITION 2.3.6. a. Ill ~ p < q < oo then W(L!.,L~,) :J L(vw)? U L(vw)?? b. Ill 5 q ~ p 5 oo then W(L!.,L~,) C L(vw)11 nL(vw)?? PROOF: Assume 1 ~ p ~ q < oo. By Propositions 2.3.2 and 2.3.4, llfllw(LJJv ' 1L ' ) ~ IQI?-? llfllwcLv' ?'L ' ) ~ IQli-? llfllL? 9 ? w' w' (1110) Thus W(L!JJ,L~,) :J L(vw)?? 73 ' ) , For the second containment we use the Minkowski integral inequality. First, write (2.3.1) llfllwcLP p ,L'l q) (L (L lf(t)I' x.q(t)v(t)? dt) ?I? w(x)? dx) ?I? II W (L (L IF(x, t)l'v(t)? dt )"' w(x)? dx )"' , where F( x, t) = f(t) ? X:i:q(t). As q/p 2:'. 1 we can apply Lemma 2.3.5, using the measures?= v(t)P dt and -v = w( x )q dx, to obtain 11/ll~(L:,L:,) = (L (L IF(x, t)l'v(t)? dt)"' w(x)? dx )"' < L( L IF(x,t)l???l?w(x)? dx )''' v(t)" dt L( L lf(;)I' X.q(t) w(x )? dx) ?I? v(t)? di Ll f(t)I' (L X,q-,(x)w(x)? dx )''' v(t)" dt. Without loss of generality, assume e E Q. Then since wq is right moderate, there exist by Theorem 2.1.6g constants C, D > 0 such that for all t E G. Thus, as desired. The remaining cases are similar, with 1 :S: q :S: p < oo following 74 by applying the Minkowski integral inequality to (2.3.1), but in the opposite direction. I The following theorem, a Holder's inequality for Wiener amalgams, can be extended to a duality theorem. However, we delay consideration of duality until after we develop equivalent discrete norms, cf., Theorem 2.5.1. PROPOSITION 2.3.7. Given 1 ~ p, q ~ oo, 1 where l + ..!.. = l + .!. = 1 v' = v 1-P , and w' = w 1-q' p p' q q' ' , , PROOF: Since (L~)' = L!, and (L~)' = L~,, we have REMARK 2.3.8. From Proposition 2.3.2, llfYllwcL1,L1) ~ ll!YIIL1, and is equality if we chose Q so that e E Q and IQI = 1. 75 Section 2.4. Discrete norms. In this section we derive equivalent discrete-type norms for the Wiener amalgam spaces, analogous to the equivalent norms (0.1.1) for the amalgams We continue to assume throughout this section that C is solid and right translation invariant. DEFINITION 2.4.1. A set of functions {'1h}ieJ on G is a bounded uniform partition of unity, or BUPU, if a. LV'i = 1, b. sup !!V'ill= < oo, c. there exists a compact set UC G (with nonempty interior) and points Yi E G such that supp(-ipi) C YiU for all i, d. for each compact KC G, sup #{i: x E YiK} = sup #{j: YiK n y3K -:p 0} < oo. zEG i We say that the BUPU has size U, and call {yi} the associated points. It has been shown in [F7] that it is possible to find BUPU's of any pre- scribed size in any homogeneous Banach space. THEOREM 2.4.2. If { v,i} is a BUPU of size U with associated points {yi}, then (2.4.1) for every compact set V :) U. 76 PROOF: For simplicity, denote the right-side norm of (2.4.1) by II? llv, This clearly is a norm, so we first show that it is independent of V in the sense that different choices of V give equivalent norms. Fix f, and let V1 , V2 :::> U be compact sets with nonempty interiors. Then we can find :z: 1 , ??? , XN such that Defining G v = 'E II f"Pi II B X y; v, we therefore have for x E G that Gv2 (x) L llf"PillB x,uv,(x) < L llf"PillB Xu1liV1:i:,.(x) i N < L llf"P,IIB LX11,V1:i:,.(x) i k=l N = LL llf"Pi IIB X11, V1 (x x; 1 ) k=l i N - L Gv1(xx;1 ) k=l N - L (Rz;iGvJ(x). k=l Since C is solid and right translation invariant, this implies ll!llv, IIGv, lie N < 11:E R:i:;iGv1 lie k=l N < L IIR:i:; 1 lie IIGv1 lie k = 1 77 = MNllfllvu where M = max { II Rz-,. 1 11 e} < oo. A symmetric argument gives the reverse inequality, so II ? II is in fact independent of V. Now we show that the left- and right-hand sides of (2.4.1) are equivalent. Fix Q large enough that u-1 U C Q. If :c E YiU then Yi E :cU-1 , so YiU C where M = sup ll1Pi\l 00 < oo. Hence, Gu(:c) - I,: 11/"Pi IIB X1uu(z) I.: 1\/1/,i \IB {i:zEy;U} < #{i: :c E YiU}M llf ? Xzql\B < Cu M \If? Xzql\B, where Cu= supzEG #{i: :c E YiU} < oo. Since C is solid, this implies (2.4.2) 11/llu = \\Guile ~ CuM 1111/ ? Xzq\\B lie = CuM 11/llw(B,e)? To prove the opposite inequality, let V :::> Ube such that V:::, UQ- 1 ? Given :c E G, define Mz {i: YiU n :cQ f= 0}. 78 If i E Mz then y,u = xq for some u E U and q E Q, so x = y,uq-1 E y, V. Therefore, iEM. iEM. iEM. ~ 1:: llf,J,.IIB X11;v(x) i = Gv(x). Since C is solid, we therefore have (2.4.3) llfllw(B,c) = 11 llf ? XzqllB lie ~ IIGvllc = llfllv- From (2.4.2), (2.4.3), and the fact that II ? llv is independent of V, we conclude that 11 ? llv ~ 11 ? llw(B,C)? I EXAMPLE 2.4.3. Assume {y.} and U are such that {y.U} is a partition of G. Then {X 11;u} is a BUPU of size U, so llfllw(L~ 1Lt) ~ II~ II!? Xv,ullL~ Xv,ullLP.. - (L IDIJ ?x ,,ullv. x.,u(xll' w(x) dx) ,,, - ( L, /.,)l ? x.,ull1: w( x) dx )"' - ( L, II!? x.,ullii f..u w(x) dx )"', 79 where the interchange of summation and integration is justified by the fact that {yiU} is a partition of G. Since w is right moderate we have by Theorem 2.1.6e, f that the values f, u w are uniformly equivalent to the values of w at 11 any point in Yiu or to its supremum or infimum on YiU. Thus, for example, where Zi E YiU is any set of fixed vectors, and w is the weight on the index set J defined by w(i) = w(z,). EXAMPLE 2.4.4. Set G = Rd, U = (0, 1] (the unit cube in R"), and Yn = n for n E zd. Then, by Example 2.4.3, the norm for W(.LP(R"), L'(R")) is equivalent to the discrete norm Thus W( LP(R ), L' (R)) is identical with the standard amalgam spaces defined in (0.1.1). EXAMPLE 2.4.5. "Dyadic amalgams" dyad(LP ,l'), considered by some au- thors, are defined by the norms e.g., (FS]. The sets {?[2n, 2n+ 1]} form a dyadic partition of R., and are group translates in R. of the compact set [1, 2] since ?[2n, 2n+I] = ?2n ?[l, 2]. However, the dyadic amalgam spaces are not Wiener amalgam spaces on the 80 multiplicative group R., because of the use of Lebesgue measure dt rather than the Haar measure dt/ltl for R.,. For example, by Example 2.4.3, a discrete IS llfllw.(L?,L?J ~ ( :E Iii? X?12?,2?+?1111.(a.J)''', nEZ,? where we recall that when dealing with the groups Rd and R: we use the notations and II II :t h 1: cf., (0.1.2) and (0.1.3). Since .. ...... .",. 11 11 ltl P u(t)IIL,ca.> - llullL,ca>, it follows that EXAMPLE 2.4.6. A d-dimensional discrete norm for the amalgam spaces W.,(LP,Lq) = W(V(R:),Lq(R:)) on the multiplicative group R: would be the following. Let G = R: and U = [I, 2] C R:. Recall the defintion fid == {-1, l}d, i.e., nd is the set of d-tuples of ?l's. Then {u2n[1, 2]}nezc1,o-enc1 is a partition of R:, where 2n = (2 n1 nd, ??? , 2 ) as usual. Therefore, 81 REMARK 2.4.7. In Example 2.4.3 we assumed that we could :find a BUPU {7 /,i} such that the supports of the -,J,i were disjoint. This may not be possible, or even desirable, in general. However, we have by definition that the supports of any BUPU { 7/,i} do not overlap "too much", i.e., s~p #{j: I supp(-,J,i) n supp(-,J,i)I > O} < oo. I This allows us to prove that W(B,Li) has an equivalent discrete-type norm based on any BUPU (Theorem 2.4.11). DEFINTION 2.4.8. A family {EihEJ of subsets of a measure space (X, ?) has a bounded number of overlaps if K - sup #{j: ?(Ei n E;) > O} < oo . i We call K the maximum number of overlaps since no E;, can intersect more than K of the E;, Note that K = II I: XE; 110()? LEMMA 2.4.9. Given a measure space (X,?) and a family {E;,}nEJ with maximum number of overlaps K. Then there is a finite partition { lr }~1 of J such that {2.4.4) i =/:- j E lr => ?(E;, n E;) = 0. PROOF: Let 1 1 be a maximal subset of J with respect to (2.4.4) for r = 1. Inductively define lr for r ~ 2 as a maximal subset of J \ U~-i J, having property (2.4.4). Suppose i E J \ Uf lr, Then given 1 $ r $ K, we have 82 i E J \ u;-1 J. and i rt. Jr. Since Jr is maximal in J \ u;-1 J. with respect to (2.4.4), JrU{i} cannot satisfy (2.4.4). That means there is ajr E Jr such that ?(Ei n E;. . ) > 0. Hence, for each l E {i,ii, ... ,jK} we have ?(Ei n Ez) > 0. However, the Jr are disjoint, so i,ii, ... ,jK are distinct, which contradicts the definition of K. Therefore J = LJf Jr. I PROPOSITION 2.4.10. Given a measure space (X,?) and given 1 ::; p::; oo. Assume {fn}nEJ C LP(X, d?) are nonnegative functions such that {supp(fn)} has a maximum of K overlaps. a. If 1 ::; p < oo then for each E.ni te set F C J we have (2.4.5) (L IIJnll:)l/p $ II L fnll $ Klfp' (L llfnll:)l/p? nEF nEF P nEF Therefore, I:llfnll~ < oo if and only ifI:fn converges in LP(X,d?). In this case the convergence is unconditional, and we can replace F by Jin (2.4.5). b. If p = oo then for each E.nite set F C J we have (2.4.6) sup llfnlloo $ II L fnll $ K sup llfnlloo? nEF nEF oo nEF Therefore, sup llfnlloo < oo if and only if I:fn converges in L00 (X,d?). In this case the convergence is unconditional, and we can replace F by J in (2.4.6). PROOF: We prove only a as bis similar. By Lemma 2.4.9, we can partitition J as J = LJf Jr, with the property that ?(supp(/m) n supp(/n)) = 0 whenever m =J- n E Jr. Recall now that for any Cn ~ 0 we have (t 1Cnp) l/p $ t Cn $ K'fp' ( t c.? )' '. 1 1 1 83 Therefore ' 11~,-11: = 11~,-rd? = / /t L f n/p d? j X r=l nEFnJ. ~ x?I?' t Ix I L ,.rd? r=l nEFnJ. N = KPIP' L L LI/ nip d? r=l nEFnJ,,. = KPIP' L //fn/1:, nEF where the next-t ll{llfV'iliB w(i)}lllq - II {I I !V'i II B} II lq wq ~ llfllw(B,L9 )? I w 9 From Proposition 2.3.4 and Corollary 2.4.12 we obtain the following. COROLLARY 2.4.13. Given 1 ~ P2 ::-; p1 ~ oo and 1 ::-; q1 ::-; q2 ::-; oo, 85 REMARK 2.4.14. Propositions 2.3.1 and 2.3.4 and Corollary 2.4.12 combine to give a simple proof of Proposition 2.3.6. For, if p ~ q then, by Propositions 2.3.1 and 2.3.4, and, by Corollary 2.4.12 and Proposition 2.3.1, 86 Section 2.5. Duality. In this section we prove, using the discrete norms obtained in Section 2.4, that W(B,C)' = W(B',C'). THEOREM 2.5.1. Given 1 :5 p, q < oo, W(L!,L~)' = W(L!:,L~,), where ?+ '?r = ?+ ;, = 1, v' = v1-p', w' = w 1-t/, and the duality is given by (f,g) = Lf (t) g(t) dt PROOF: We assume for simplicity that {X11iu} is a BUPU for G, so by The- orem 2.4.11, (2.5.1) and (2.5.2) where Zi E YiU is any fixed choice of vectors and w(i) = w(z,), w'(i) w'(z,). The case of a genera.I BUPU {?,} is similar, with some added technical complications. I I a. Given f E W(L!,L?i,) and g E W(L!,,L~,) we have Ll f(t) g(t)I dt = L J.,;U lf(t) g(t)I dt < L Iii? x.,,ullLt lln ? x.,iullL,.,', 87 Therefore f Ofg is well-defined, and so g determines a continuous linear functional on W(~,L~). b. We show now that IIYllwcL::,L~,) = sup{l(f,g)I: llfllwcL!,Li) = 1}. I I To see this, assume for simplicity that 1 < p, q < oo, fix g E W( L!,, L!, ), and define 9i = 9 ? X11,u. Let fi(t) = { l9i(t)IP' v'(t) / 9i(t), 9i(t) # O, o, 9i(t) = 0. Then supp(J.) C Yiu, and lh(t)IP v(t) = l9i(t)lp(p'-l) v(t)P(l-p') v(t) = lg,(t)lp' v'(t), I so IIJ.lli11? = IIYillJ>L 1.1,, < oo. Moreover, (2.5.3) (f.,gi) = LJ.(t)gi(t)dt = Ll 9i(t)jP' v'(t) dt (L lg;(t)i'' v'( t) dt )''' (L lg,(t)i'' v'(t) dt )'''' 88 = (J /o )1/p (1 )1/p' 1/i(t)IP v(t) dt G l9i(t)IP' v'(t) dt Define Ci = { b/ w'(i) / (ai bi), aibi :/= O, 0, a,b, = 0. Then ( Ci ai)q w(i) = b,q(q' -l) w(i)q(l-q') w(i) = b/ w'(i), (2.5.4) Note that (2.5.5) and define f = Z: Ci Iii this is possible as {X,,u} is a BUPU. We have 89 so f E W(L~, Lt). And, from (2.5.3), (2.5.4), (2.5.5), and (2.5.6) we have which completes the claim. c. Finally, assume that ?, E W(L~, Lt)' is given. Fix i, and note that {h E L~(G): supp(h) C YiU}, since, by (2.5.1), Therefore ?, restricted to L~(yiU), defines a continuous linear functional on I L~(yiU), so there exists a 9i E L~(yiU)' = L:,(yiU) such that (h,?,) = (h,gi) for h E L~(yiU). Since supp(gi) C Yiu and {yiU} is a partition of G, we can define g = ~ 9i? To show that g E W(L:'.,L~,), we first claim that {ll9illL.., '} E Lt,. Given , {ci} E ?i and e > 0, choose /i E L~(yiU) such that llfillL! ~ 1 and Note that f = ~cdi E W(L~,Li) since 1/q ll/llwcL~,L:,,) ( L llcddl1! w(i) ) < 90 Hence, (2.5.6) \Lci(/i,gi)\ = II:ci(fi,?)I = \(I:cdi, ?)I = \(!,?)I Without loss of generality, fix the phase of Ci so that Ci (/i,gi) ~ 0. Then, using (2.5.6), L lei\ \l9i IIL:: '.5 L lcil (1(/i, 9i) \ + 2i\ci\) = \L ci(fi,gi) \ + e ~ II {C i} II lq.., ll?\I + e . , Thus { ll9i I\ LP' } E (it)' = f~,, as claimed. Hence, v' < oo, I I so g E W(L!,,L~,). Clearly (/,g) =(!,?)for all/ E W(~,L~), so we are done. I 91 PART III GENERALIZED HARMONIC ANALYSIS 92 CHAPTER 3 BESICOVITCH SPACES In this chapter we establish the basic properties of the Besicovitch spaces B(p, q). These space were defined, for the one-dimensional case, in (0.2.14); the general definition is given in Definition 3.2.1. Our main result, Theorem 3.2.4, is that B(p, q) coincides with the Wiener amalgam space W.(LP, Lq), where we recall that for notational simplicity, and to avoid confusion between amalgams on the additive and multiplicative groups, we adopted the notations and cf., (0.1.2) and (0.1.3). Our identification of B(p, q) as W.(LP, Lq) immediately provides us with equivalent discrete norms for B(p, q), and implies duality and inclusion re- lations. These basic properties provide the machinery for our results on the Wiener transform in Chapter 4. Although not pursued in this thesis, the Wiener space identification implies other properties as well, e.g., convolution relations on the multiplicative group. We begin in Section 3.1 by considering higher-dimensional analogues of the nonlinear spaces B(p,lim) defined, for one-dimension, by (0.2.4). We review 93 the definitions of higher-dimensional limits from [BBE] (needed to define B(p, lim)) and prove the nonlinearity of B(p, lim) in higher dimensions. In Section 3.2 we prove the fundamental equality B(p, q) = W*(LP, Lq) and establish bounds for the norm equivalence. We do this in terms of the discrete norm for W*(LP, Lq), as it is this norm that we use to prove our results in later chapters. We also discuss the inclusion and duality relations that follow from this identification. In Section 3.3 we prove a higher-dimensional analogue of a theorem due to Beurling, which characterizes B(p, oo) as an intersection of weighted ?P_ spaces. We give Beurling's proof, for d = 1, and two new proofs for d ~ 1. One proof uses the Wiener amalgam norms, and is generalized in the following section to a larger class of spaces, while the other proof is valid only for B(p, oo). In Section 3.4 we attempt to characterize B(p, q) as an appropriate union or intersection of weighted LP-spaces. This reveals links between B(p, q) and other function spaces which have arisen in harmonic analysis. Finally, in Section 3.5 we examine the effect of replacing the factors 1/IRTI in the definition of B(p, q) by general functions p(T). We show that the result- ing spaces B p(p, q) are weighted Wiener amalgam spaces on the multiplicative group. 94 ~ DBS 2 - ..AZSC-::::is.ZZ:x jp! Section 3.1. Rectangular limits. The paper [BBE) extended the Wiener-Plancherel formula (0.2.3) to higher dimensions. The higher-dimensional version, (0.2.18), requires the use of special d-dimensional rectangular limits. It is the purpose of this section to define these rectangular limits, and to show that the spaces B(p, lim), consisting of functions for which the left-hand limit of (0.2.18) exists, are nonlinear, and therefore not conducive to the usual methods of functional analysis. DEFINITION 3.1.1 [BBE]. Given a function f: Rd~ C and given z EC. a. We write lim,. .... 00 f(t) = z if limreB.,r--too f(rc) = z for every c E Sd-1 \Ad. That is, f(t) converges to z along every ray from the origin to infinity except for those rays which lie in the coordinate hyperplanes. b. We write Glim,. .... 00 f( t) = z if for ea.ch e > 0 there exists a T E Ri such that lz- f(t)I < e for all t r/:. RT? This is the natural definition of convergence for Rd considered as a locally compact group, and indicates convergence to z along every path whose points are eventually arbitrarily far from the origin. c. We write Ulim,. .... 00 f (t ) = z if for each e > 0 there exists a T ER t such that jz - f(t)I < e for all t E Rd such that jt;j > T; for each j. The letter U stands for "unrestricted"; this notion plays a role in multi-dimensional Fourier series, cf., [A; Zy). We make corresponding definitions for the limits as t ~ O, and make the obvious adjustments for f defined only on Rt. For real-valued f we allow 95 z = ?oo. It is clear that if Glimf(t) exists then Ulimf(t) will exist also, and if Ulim f(t) exists then limf(t) exists also. In one dimension, the three limits are identical. The following example shows they are distinct for d ~ 2. EXAMPLE 3.1.2. Parts a and bare from [BBE]. a. Set d = 2 and f = Xs, where E {(u,v) ER~: 0 < v < u 112}. Given c E S1 we have limrER,r-+oo f(rc) = O, so limt-+oo J(t) exists and is zero. However, Ulimt-+oo f(t) does not exist. In fact, given T E Rt we can finds, t E R 2 withs, t > T such that f(s) = 0 while J(t) = 1. b. Set f = Xs where E { t E Rd : It j I > 1 for all j}. Then Ulimt-+oo J(t) = 1 although Glimt-+oo J(t) does not exist, c. Set f = Xs, where E {t E Rd: 0 < t1 < 1}. Then Ulimt-+oo J(t) = 0 although Glimt-+oo J(t) does not exist. DEFINITION 3.1.3. Given f E Lf c(Rd) and a set E C Rd with finite mea- 0 sure, the mean off on E is Ms(f) l~I Lf (t) dt. 96 Hit exists, the (rectangular) mean off is EXAMPLE 3.1.4. We give examples of functions which do or do not possess means. One-dimensional versions of parts d and e appeared in [Bal], of part fin [HW], and of part gin [LL]. a. If f E L1 (Rd) then M(f) exists and is zero. For, given TE Ri we have b. If f E Lf c(Rd) is P-periodic, where PE Ri, then 0 M(f) = l~I 1f (t) dt, where IC Rd is any rectangle with side lengths P. To see this, fix TE Ri, and let N = N(T) E Zi be the unique vector such that NP~ T < (N + I)P. Note that since f is P-periodic. Therefore, 1 1 1 1- f- -III 11f \RTI RT I 97 IR(N+l)PI - IRNPI IR(N+i)PI 1 ( Iii IRNPI IRNPI IR(N+i)PI JR(N+1)P + IRcN+l)PI 1 r 111 1 r 111 IRNPI IR(N+l)PI JR(N+1)P IRNPI JRNP - C IR(N+l)PI - IRNPI IR(N+l)PI + C IR(N+i)PI C IRNPI IRNPI IRNPI C II(N + 1) - II{N) II{N + 1) + C II(N + 1) - II(N) II( N) II(N ) II( N) ' where c = mIr 111. Since Glim II(N + 1) - II(N) = 0 and Glim II(N + l) _ 1, N-+oo II(N) N-+oo IT(N) the result follows. c. From part b and the fact that Eb(t) = e21rib?t is 1/b-periodic, we have M(Eb) = III(b)I r Eb = Dob? J(o,1/b] d. The function J(t) = III(t)I does not have a mean, since liTI JRT f = III(T/ 2)1. e. The function J(t) = III(t)li is bounded, yet does not possess a mean, since iixl fRx J = III(T)li /(i + l)d. Note, however, that M(IJIP) does exist for all p > 0 since If I = 1. 98 f. By part b, any / E Lf0 c(Rd) which is periodic possesses a mean, even though it need not be bounded. All bounded periodic functions possess means. g. Let { tn}neZ be any sequence of positive real numbers strictly increasing to infinity which satisfies - 0. Set t 0 = 0 and let Then the function f = 'XE does not possess a mean, despite the fact that it is bounded and takes only the values O and 1. To see this, fix c E S!_1 and define Tn = tnc, Note that RT1 C RT2 C ... , and \RT. . \ = (2tn)d \II(c)\. Therefore, -+ 0 as n -+ co. However, 1 / / > 1 /, l \RT:,,.+1 \ j RT 2n+i IRT2n+i \ RT:,,.+i \RT2,. IRT2 .. +il - \RT2 .. I IRT2 .. +1I 99 -1- 1 as n -1- oo. Therefore M(f) does not exist. Also, M(IJIP) does not exist for any p since Iii= f. h. Let f be as in part g, and set h = l /2 and g = f - h. Since M is linear and M(h) exists while M(f) does not, we conclude that M(g) does not exist. However, IYI = 1/2, so M(jglP) exists for all p, even though M(IJIP) does not. EXAMPLE 3.1.5. We show that B(p,lim) = {f E Lf c(Rd) : lim - l I / Jf(t)IP dt exists} 0 T-+oo 1R T }RT is nonlinear. a. Let f, g, and h be as in Example 3.1.4g. Since !YI = lhl = 1/2 we have g, h E B(p,lim). However, M(f) = M(IJIP) does not exist, so f = g + h 1 B(p,lim). b. For p = 2 we give another example, whose one-dimensional version appeared in [Bal] and [HW]. Let f E B(2,lim) be any function such that M(f) does not exist (see Example 3.1.4e for a complex-valued example, or Example 3.1.4g for a real- valued example). Given any TE Ri we have Now, M(l) and M(l/12 ) both exist while M(f) does not. Therefore 1 + f 1 B(2,lim) even though 1, f E B(2,lim). 100 Although nonlinear, B(p, lim) is a large space. For example, it contains V(Rd) and all periodic functions which are integrable over their periods, including all constant functions. Examples 3.1.4e and f show that B(p, lim) \ L 00 (Rd) -f- 0 and L 00 (Rd) \ B(p,lim) -::/- 0. REMARK 3.1.6. The original Wiener-Plancherel formula, (0.2.3), was proved by Wiener for functions in B(p,lim), in one dimension. Because B(p,lim) is nonlinear, Lau and his colleagues extended the Wiener transform to larger spaces. In [LL], where they proved that B(p,lim) is nonlinear, Lau and Lee proved ( also for d = 1) that the Wiener transform W is a topological iso- morphism of the Marcinkiewicz space B(2,limsup) onto the variation space V(2,limsup), where B(p,limsup) and V(p,limsup) are as defined in (0.2.5) and (0.2.6), respectively. Of course, B(2, lim sup) ::> B(2, lim), and, by the Wiener-Plancherel formula, the Wiener transform is an isometry when re- stricted to B(2,lim). However, Lau and Lee proved that Wis not an isometry on all of B(2,limsup), not even on the linear span of B(2,lim) in B(2,limsup). Following the Lau and Lee results in [LL], Lau and Chen proved in [CLI] that it is also possible to extend the Wiener transform W from B(2, lim) to B(2,oo), and that Wis a topological isomorphism of B(2,oo) onto V(2,oo), where B(p, oo) and V(p, oo) are as defined in (0.2.7) and (0.2.8), respectively. This result forms one cornerstone for our results in Chapter 4, for we prove there that W is in fact a topological isomorphism of a whole range of spaces B ( 2, q) onto V ( 2, q) for 1 ~ q ~ oo. Moreover, we do this in higher dimensions. 101 A goal for future research is to extend the Lau and Lee results for B(2, lim sup) to higher dimensions as well. As a step in this direction, we make a few remarks on the definition of d-dimensional rectangular limsups. DEFINITION 3.1.7. Given a real-valued function f:Rd--+ R. a. limsupt-+oo J(t) = supcESc1_ 1 limsuprER,r-+oo J(rc). b. Glimsupt-+oo J(t) = infTeRt supttRx f(t). c. Ulimsupt-+oo J(t) = infTeRt suptER",lt; l>T; f(t). We make corresponding definitions for liminfs, for t --+ O, for f: Ri --+ R, etc. Note that the numbers defined above always exist in the extended real sense, i.e., -oo::; limsupf ~ oo. Given J:Rd--+ R we have Gliminf f < Uliminf f < liminf f < limsup f < Ulimsup f < Glimsup J. However, these are not equalities in general, cf., Example 3.1.8. Also, it is clear that lim J(t) exists {} liminf J(t) = limsup J(t), t-+OO t-+oo t-+oo Glim J(t) exists {} Gliminf J(t) = Glimsup J(t), t-+oo t-+OO t-+OO Ulim J(t) exists {} Uliminf J(t) = Ulimsup f(t). t-+oo t-+oo t-+oo EXAMPLE 3.1.8. a. Let f be as in Example 3.1.2a. Then liminf f = limsupf = O, Uliminff = O, Ulimsupf = 1, Gliminff = O, Glimsupf = 1. 102 - - - - ~~~-4-.. j.:; .... - -_-..,.._-.;:.,..-,.,.-_ ___ '_,e::. __ b. Let f be as in Example 3.1.2b. Then liminf f = limsupf - 1, Uliminff = Ulimsupf = 1, Gliminff = O, Glimsupf = 1. c. Let f be as in Example 3. 1.2c. Then lim inf f = lim sup f - 0, Uliminf/ = Ulimsupf = o, Gliminff = O, Glimsupf = 1. 103 Section 3.2. Equivalence with Wiener amalgam spaces. In [F4), Feichtinger derived an equivalent norm for B(p, oo) based on dyadic decompositions of R (in fact, this was done in higher dimensions, but with a spherical approach, rather than the rectangular approach of this thesis). Essentially, he proved that B(p, oo) = W.(LP, L 00 ), under equivalent norms. We prove and extend this equality in this section, namely, we show that B(p, q) = W.(LP, Lq) for all p, q, and do this in higher dimensions with a rectangular approach. We adopt the discrete norm for W.(LP,Lq) defined in Example 2.4.6 as standard, i.e., we take {Xo-[2 n 12 ,.+1]}nEzci,o-EOci as a standard BUPU, with the result that / dt )q/p)l/q (3.2.1) llfllw.(LP,Lq) = ( L ( },. n n+l lf(t)IP IIT(t)I ' nEZd ,o-EOd u(2 ,2 ] the standard adjustments being made if p or q is infinity. DEFINITION 3.2.1. Given 1:::; p, q < oo, the Besicovitch space B(p,q) is the space of functions f: Rd----+ C for which the norm 1 / ( 1 / ) q IP dT ) / q 11!11B(p,q) = ( }Rt IRTI }RT lf(t)IP dt IT(T) is finite. The standard adjustments are made if p or q is infinity, namely, IIJIIB(p,oo) 1 1 )l /p ess sup ( -\RI lf(t)IP dt , TERt T RT llfllB(oo,q) (Lt (?:~:~p[f(t)1r n~~if ?, 104 llfllB(oo,oo) = ess sup ( ess sup IJ(t)I). TEB.i tERT That II ? IIB(p,q) is a norm is evident. It follows from Theorem 3.2.4 that B(p, q) is a Banach space. Our characterization of B(p, q) as a Wiener amalgam space begins with the easiest case, namely, p = q. PROPOSITION 3.2.2. Given 1 $ p $ oo, with PROOF: The case p = oo is clear, so assume 1 $ p < oo. The second equality is trivia.I, since llfllw.(LP,LP) (L (1 lf(t)IP ~)p/p)l/p n,u o-(2" ,2"+1] III( t) I (L,. IJ(t)IP 1n~:i1) ?IP = ll!IILP? For the first equality, compute 105 LEMMA 3.2.3. Given a E Rt. a. L rr(2-ka) = rr(2a 1_ 1)? kEZi b. L rr(2-(k-l)a) = rr(2}~1)? kEZi PROOF: We compute L ... L 2-k1a1 ... 2-kdad kdEZ+ k1EZ+ d II L 2-k;a; j=l k;EZ+ d 1 - II 2a; -1 ? j=l The second statement is similar. I The following is the main result of this chapter, in which we characterize B(p, q) as a Wiener amalgam space. The bounds given for the norm equiva- lence in Theorem 3.2.4 are not sharp, cf., Remark 3.2.5. THEOREM 3.2.4. Given 1 ~ p < oo and 1 ~ q ~ oo, 106 with equivalence of norms given by (3.2.2) C II? llw.(LP,Lq) :S 11 ? IIB(p,q) < D 11 ? llw.(LP,Lq), where p < q, p '2:. q. PROOF: Assume for simplicity that 1 :Sp, q < oo (the q = oo case is similar). a. Fix any u E nd. Then we compute / ( 1 / )q/p dT llflliJ(p,q) = lat !RT! }RT IJ(t)jP dt IT(T) = L 1 ( 1 / IJ(t)IP dt)q/p __!!__ n [2",2n+l) 2d Il(T) }RT II(T) where the summations inn run over zd. Therefore, > (log 2)d 2-ldq/p L (1 IJ(t)jP __!!!:._) q/p n,u u[2n-1,2nJ jTI(t)j 107 = (log 2)d 2-3dq/p IIJll~.(L?,L?)' from which the first inequality in (3.2.2) follows. b. Note that u Therefore, 11111i(p,q) = L/ ,;T, 1, \f(t)\? dr? II~~) = L r ( 1 r lt(t)IP dt)q/p dT n J[2n,2n+1] 2d IT(T) )RT II(T) < L r ( 1 r 11(t)1P dt)q/p _!!__ - n 112n ,2"+1) 2d II(2n) JR 2 .. +1 II(T) = ~ (log 2)? (11(2~+1) L, .? ,\ f(t)\' dtr ? = 1 1 )q/p (log 2)d L ( II( n+l) L IJ(t)IP dt n 2 m,T T[2"-m+1,2 ,.-m+J] n-m+2) 1 dt ) q/p < II(2(log 2)d - ~ ( ~ IJ(t)IP --II(2n+l) T[2"-m+1,2 .. -m+2] III(t)I = (log 2)d L 1 1 dt )q/p ( ~ t 1/(t)IP --n II(2m- ) T[2"-m+1,2,.-mH] IIT(t)I q/p = (log 2)d L IL Fm,T(n) n m ,T l = (log2)d !IL Fm,T11::;P, ffl 1T 108 where Fm,r is the sequence Fm,r(n) = 1 1 m 1 lf(t)IP -d-t . Il(2 - ) r[2n--+1,2n- ... +2] jll(t)I c. Assume p :5 q, i.e., q/p 2:: 1. Then we may apply Minkowski's inequality in the Banach space l,qf p to the calculation in part b. The summations in the following calculation are over m E Zi, n E zd, and u, -r E nd. m,r m,r p/q ~ ( ~ \Fm,r(n)\qfp ) m,r n ~ (~ (rr(2:-,l f.12?--+',2?--+>1 l/(t)I' III~:lr?r? - ~ IJ(2:_, l (~ (!.,,.,,.+,] l/(t)I' 1~!iir'T'' < L II(2:-1) (~ ( J/ ,. ?+1 lf(t)IP IITd(!)1)q/p)p/q m,r n,.,. 0'[2 ,2 ] since the summation in mis over Zi, The second inequality in (3.2.2) there- fore follows for this case. d. F inally, assume q :5 p. Since O < q/p :5 1, we may apply the triangle in- equality in the metric space l,q/p to the calculation in part b ( cf., Section 1.7f). 109 The summations in the following calculation are over m E Zi, n E zd, and ni,r ni,r rn,r n where the last equality follows from Lemma 3.2.3 and the fact that the sum- mation in m is over Zi. The second inequality in (3.2.2) therefore follows for this case. I REMARK 3.2.5. The bounds for the norm equivalence given in Theorem 3.2.4 are, in general, not the best possible. For example, for the case p = q we can compare the exact bounds deter- 110 mined in Proposition 3.2.2, namely, to the approximate bounds given in Theorem 3.2.4, i.e., (2-c log2)'IP 11 ? llw.cL,,LP) $ II? lls(p,p) ~ (2 log2)"/P 11 ? /lw.(L',L')? Since we conclude that the bounds in Theorem 3.2.4 are not best possible. REMARK 3.2.6. Our recognition of B(p,q) as the Wiener space W,..(LP,?9 ) immediately provides us with inclusion and duality relations. a. Incl.usion.s. From Corollary 2.4.13, From Proposition 2.3.2 ( cf., Proposition 3.2.2), B(p,p) = W,..(LP,?P) = ?P(R!). Therefore, and cf., Proposition 2.3.6. 111 b. Dilation invariance. By Proposition 2.2.8, W..,(LP,Lq) is dilation invari- . ~ ch ). E Rd where D>.. is the ant, i.e., IID>../llw.(LP,Lt) ~ llfllw.(L?,L') ior ea. "'' dilation operator D>.J(t) = f(t/>.). In fa.ct, B(p, q) is dilation isometric, since 1 I )q/p dT IID?flltcp,q) - J.." ( IRxl la-r lf(t/>.)IP dt II(T) + dt )q/p dT - J.." c;TI L,.,r lf(t)jP III(>.)I II(T) + t / )q/p dT - L_.,. ( IR?TI lau~ lf(t)IP dt II(T) + Ii ( t 1 )q/p dT - - 1/(t)IP dt II(T) R" IRxl R-r + c. Duality. From Theorem 2.5.1, if 1 < p, q < oo then B(p,q)' - B(p',q'), with duality given by = f - dt (/,g) J. ." J(t)g(t) III(t)I" ? Since the norm in B(p, q) is only equivalent to the norm in W..,(V, Lq), we can conclude only that the norm in B(p, q)' is equivalent to the norm in B(p', q'). The following computation shows that the canonical norm for B(p, q) ' is a. consta.nt multiple of the norm for B(p' ,q'). Given f E B(p,q) and g E B(p'' q'), 112 IIJIIB(p,q) ll9IIB(p',q') (Lt c~TI L, 11 t c~I JR, 1/(t)I? dtf ?c ~TI L, lg(t)1?? dr?? IT~~) > Lt l~TI L lf(t)g(t)ldt IT~~) llfgllB(l,1) > 2-d IL: f(t) g(t) 1rr~:)1 I - 2-d 1(1,g)I, The norm for B(p, q)' would be equal to the norm for B(p', q') if we defined the duality by fl{ -dT (f,g) = JRt IRTI }RT f(t)g(t) dt IT(T)' i.e., duality according to the norm for B(l, 1). REMARK 3.2.7. Although the sets W.(LP,Lq) and B(p,q) coincide by The- orem 3.2.4, they have distinct, albeit equivalent, norms. We retain this dis- tinction in the remainder of this thesis, stating results in terms of W.(LP, Lq) when we intend to use the discrete norm, or in terms of B(p, q) when we intend to use the norm for that space. If the norm is not important, we refer to the space as B(p, q). 113 We close this section with a few remarks about the space B(p, oo ). LEMMA 3.2.8. Given O < p < oo and b ER~, we have Eb - 1 E B(p,lim) C B(p,oo), with PROOF: Without loss of generality assume b E Ri, Since Eb - 1 is 1/b- periodic and bounded it is an element of B(p, lim) by Example 3.1.4b. More- over, that example also implies that llEb - 11\~( ) 2: lim IRl I / IEb(t) - lip dt p,oo T--+oo T j RT 1 _ I[ /bll r IEb(t) - 11P dt 0, 1 J[0,1/b] r IE1(t) -11P dt J[o,1] - IT 11 le21rit; - 1 jP dt; j=l 0 > ITd 13/4 lehit; - lip dt; j=l 1/4 d (3/4 2: !! 11/4 ( v'2Y dt; = 2 2d(?-}) > 0, 114 ------------? if a :/: b. Thus { EbheR" is an uncountable separated set in B(p, oo ). I The statement and proof of the following result is adapted from the one- dimensional version presented in [Bal]. P1toP0s1TION 3.2.10. Given 1 ~ p < oo, B(p,lim) is a. proper, closed, non- linea.r subset of B(p,oo). Moreover, if {ln}nEZ+ C B(p,lim), IE B(p,oo), a.nd In-+ I in B(p, oo), then IE B(p,lim) and M(/1/P) = nl-+imOO M(lfn/P), where M is the mean value opera.tor of Definition 3.1.3. P1tooF: Clearly B(p, lim) c B(p, oo ), and is nonlinear by Example 3.1.5. In 00Example 3.1.4 we showed that B(p,lim) \ L00(Rd) f= 0 and L (Rd) \ B(p, lim) -f: 0. Since both B(p, lim) and L00(Rd) are contained in B(p, oo ), B(p, lim) must be a proper subset of B(p, 00) . Now assume that In E B(p,lim) and In ~ I E B(p,oo). Set Mn - M(lfn/P), and note that M _ M / < lim MR2'(/lm - ln/P) I m n - T-+oo < sup MR2' (/Im - ln/P) - TERt - I/Im - ln/l~(p,oo) -+ O as m, n ~ oo, where MRT is the mean value on RT- Mn must therefore converge to some number M as n ~ oo. 115 ?? .a -- By hypothesis, P) - llf - f nll~(n,00) --+ O as n-+ 00, ?n = sup MR-r ( If - f n I r Teat Therefore, for T E Ri, Thus, < limsup MRx(l/1"') - T-oo < limsup MRx(lfnl"') + ?n - T-oo - Mn + en, where the liminfs and limsups are the d-dimensional versions defined in Sec- tion 3.1. Letting n-+ oo, it follows that M(lfl"') = limT-00 MR-r (Iii"') exists EXAMPLE 3. 2 .11. Hartman and Wintner [HW] gave the following example (for d = 1) of functions {fn}neZ+ C B(2,lim) n L00(R) and f E B(2,lim) such that M(lf - fnl 2 ) -t O as n-+ oo but f ~ L00 (R). 116 Fix any f E L2 [0, 1)\L00 [0, 1), and extend f periodically to R. Then f E B(2,lim) by Example 3.1.4b. Let SN be the N th partial sum of the Fourier series of f, i.e., N SN(t) - L Cne2,rint, n=-N where Cn = fol f(t) e-2,rint dt. Clearly SN E B(2,lim) n L00 (R), and, by Example 3.1.4b, --+ 0 as N --+ oo. This example extends trivially to higher dimensions as follows. Let f, SN be as above, and define g(t) = f(ti) and TN(t) = SN(t 1 ) fort E Rd. Then N --+ oo. 117 Section 3.3. Beurling's characterization of B(p, oo ). Wiener, in [Wl], proved that B(p,oo) is contained in a certain weighted LP space (for d = 1). This result has been generalized by Beurling, Lau, Benedetto, and others, and we generalize it in Section 3.4 to the B(p, q) spaces. In this section, we discuss the B(p, oo) case. We begin by proving and extending Wiener's result, which in its original form is the following theorem with d = 1 and a = 2. Lau and Lee generalized this to d = 1, a > 1 in [LL]. Benedetto, Benke, and Evans proved a d ~ 1, a = 2 result in [BBE]. Our proof is a combination of the [LL) and [BBE] results. The proof is essentially Wiener's, i.e., integration by parts. THEOREM 3.3.1. Given 1 :5 p < oo and a E Ri with a> 1, B(p, oo) C L!(Rd ) where d 1 v(t) = }] 1 + jt;ja;. Moreover, the containment is proper. PROOF: a. For clarity in proving the containment we restrict ourselves to d = 2 (the general case being similar). Fix a, b > 0 and/ E B(p, oo ). Define O 8 We compute: (3.3.1) /T 1s ip(x, y) dx d Jo O (1 + x")(l + yb) y = IT Ls 1 1 h 8z 811 ,p(x, y) dx dy Jo O 1 + x" 1 + y - 1T 1 + (ls 1 + az (8,,f,(x, y)) dx) dy o 1 yh o 1 z" 1T 1 (811,p+ .....+;.(.S, ..y-)- + ls (lz+a -..1a . a )l ) 8 1/J(x,y)dx dy 1 yh 1 Sa O ... 11 Before estimating [1 and [ 2 , note that IT 1 Jo l+yh8111/J(x,y)dy and that yh-1 1 and -l+~if < -y for all z, y > 0. Therefore, (3.3.2) 1 (1/J(S 1 T) IT yh-1 ) - l+S" 1+ Th + b ) (l+yh)21/J(S,y)dy 0 119 < S T 'if,(S, T) 1 + sa 1 + Tb ST bS IT 1 'if,(S, y) + 1 + sa lo 1 + yb Sy dy < s T M s bM IT 1 d l + Sa 1 + Tb + 1 + Sa j l + yb y. 0 Similarly, (3.3.3) I,(S, T) a t (l ~::)' ([ 1 : y' &,,J,(x, y) dy) dz 1s xa-1 ('if,(x, T) 1T yb-1 ) = a lo (1 + xa)2 1 + Tb + b o (1 + yb)2 'if,(x,y)dy dx < a T 1s 1 'if,(x, T) dx 1 + Tb O 1 + xa xT + 5 ab l IT 1 1 1/J(x,y) d dz lo lo 1 + za 1 + yb xy y < T aM1s 1 dx 1 + Tb O 1 + xa + abM Is 1 dx 1T 1 b dy. Jo 1 + z" o 1 + y Combining (3.3.1), (3.3.2), and (3.3.3), letting S, T ~ oo, and noting that lims-+oo S/(1 + sa) = limT-+oo T/(1 +Tb)= o, we obtain (3.3.4) llfllit = 1-:1-: lf(z,y)IPv(z,y)dxdy -100 loo cp(z,y) dxd - 00 Jo (l+z )(l+yb) y ~ abC(a)C(b)M = 4abC(a) C(b) llfll~(p,oo)' 120 where 00 = / 1 00 C(r) Jo dx < f r 1 dz + 1 z-r dz _ 1 + zr - Jo < 00, r-1 This completes prooi oi the containment. b. To see that the containment is proper, first assume that d = 1 and define tCa-1)/(2p} t > 1 f(t) = { o, ' - ' t < 1. Then r lf(t)IP dt = roo t I = 1.- k(t) dt. 123 L ? al "'-k is so even and integrable, and 1?s d ecreas?m g on R +? Since k* is nonnega- tive, but not positive, k* is not an element of A(R). Note that~= k?X[o,1/2), so f."" k.,( t) dt = J.' I? k( t) dt "' 0.903 < 1 - f."" k(t) dt. c. For arbitrary d, define fort E Rd. Let K* be the least decreasing majorant of Kon Ri and K* the greatest decreasing minorant of k on R'.f.. That is, fort E Ri, d K*(t) sup K(s) - IT k*(t;) ,e{t,00) j=l and d K*(t) = inf K(s) = IT k*(t;). ?E{O,tj . 1=1 Extend K* and K* evenly to Rd. It follows from part b that K* E A(Rd) The functions k and K play an important role in Chapter 4. LEMMA 3.3.5. Given a nonnegative, even function w on Rd which is decrea.s- m. gon R+d . a. supTea.t II(T) w(T) < fa., w(t) dt. + b. supTEB.i II(T)w(T) < 2d supnEZ-' II(2n)w(2n). c. fa.t w(t) dt :5 Enez-' II(2n)w(2") :5 2d fa,. w(t) dt. + 124 - d. limT-0,00 Il(T) w(T) = 0. PROOF: a. As w is decreasing on Ri, ( w(t)dt 2 f w(t)dt 2 ( w(T)dt = II(T)w(T). lat Jco,TJ lco,TJ b. If TE [2n, 2n+1J then II(T)w(T) :'.5 I1(2n+l )w(2n) since w is decreasing. c. Since w is decreasing on Ri, f w(t)dt = I: / w(t)dt lad. lc2?,2?+11 + < L f w(2n)dt Jc2 .. ,2 .. +11 _ I: II(2n)w(2n) = L 24 f w(2n)dt Jr.,, .. -1 ,2" i _ 24 f w(t) dt. lat since w is decreasing on R+. This, combined with part c, gives the result. I The spaces AP, BP defined below are rectangular analogues of the spherical spaces defined in [Bel]. DEFINITION 3.3.6. Given 1 :'.5 p < oo. a. BP= nweA ~(R4 ), with norm 1 JIfJ 1/(t)IP w(t) dt ) tP sup llf llL~ = sup Rdl UvwEHAi?,1 wEA ( W ( t )dt R" 125 b. AP' - u LP' ( d - weA w' R ), wherep. !+ p-1 ; = 1 and w' -- w 1-P' , wi"th norm /If/IA,' == inf I/Ill ,, WEA, LI f/w/11=1 "' = (L 1 lf(t)/P w'(t)dt) 1/p'(L w(t)dt) 1/p inf . wEA R" R" Note that Boo = Loo(Rd) and Al= Ll(Rd). REMARK 3 .3. 7. The following facts are proved by Beurling in [Bel] (in a spherical setting). a. AP , BP are Ba nach spaces. c. AP C L1(Rd) and is a convolution algebra. d. (AP)'== BP', under the duality (f,g) = JR,, f(t)g(t)dt. e. BP== B(p ,oo ) . Each of these facts is proved in this thesis, some using Beurling's methods, some following from other results. We prove that BP = B(p, oo) in Theorem 3?3?9, from which it follows that BP is a Banach space and contains L00(Rd). We Prove the dualty (AP)'= BP' in Proposition 3.3.10, and as an immediate corollary obtain that AP = ll(t)B(p, 1), from which it follows that AP is complete and is contained in Ll(Rd). We prove in Proposition 3.3.13 that AP lS a convolution algebra. In Theorem 3.3.9 we prove that BP =B (p, oo ). The key fact is given by the following proposition, for which we give three proofs. First, we give Beurling's 126 proof for the case d = I. This proof is essentially Wiener's integration by parts technique, adapted to cover general w E A(R) by using Riemann- Stieltjes integration ( the technique can be extended to higher dimensions). Next we present a proof ford~ l which uses a method suggested to us by C. Neugebauer, who credited it to R. Bagby [Bag]. Finally, we present a simple proof based on discrete norm techniques, which proves the result but with bounds inferior to those obtained by the other techniques. PROPOSITION 3.3.8. Given a rectangular, nonnegative, integrable, decreas- ing function w 011 Ri, i.e., assume there exist w;:R+ -+ [O,oo) which are integrable and deci?easfog Oll R+, such that w(t) = n: w;(t;) fort E Ri. Then for any nonnegative cp on Ri, f ip(t)w(t) dt ~ ( f 1 w(t) dt) sup - ( f cp(t) dt. lat lat Teat II T) 110,T] PROOF: Set M = SUPTeRi nm lro,T] cp(t) dt and assume without loss of generality that M < oo. a. We begin with Beurling's proof, ford= 1. Define ,p(T) = 1T ip(t) dt; then {3.3.6) 1 M = sup T ,p(T) < oo. T>O As 'P E Lf0 c(R), ,P is locally absolutely continuous on R. Since w is decreas- ing, positive, and integrable on {O, oo) it is of bounded variation on each finite 127 closed interval [a, b] c (0, 00 ). Therefore the Riemann-Stieltjes integral 1bw (t) d-,/,(t) = Lb ip(t) w(t) dt 41 exists. Further, integration by parts, equation (3.3.6), and the fact that w is decreasing gives (3.3.7) 1b w(t)d-ip(t) _ w(b),t,(b) - w(a),t,(a) - Lb ,t,(t)dw(t) < w(b)-ip(b) - w(a)-ip(a) - 1b Mtdw(t) - w(b)-,/,(b) - w(a),t,(a) - M(bw(b) - aw(a) - Lb w(t)dt). By Lemma. 3.3.5, limT-o,oo Tw(T) = 0. Therefore, by (3.3.6), we have limT-+0,00 -,/,(T) w(T) = 0 as well. Applying this to (3.3. 7), J.~ ,p(t)w(t)dt = ~!, { w(t)d,fo(t) $ M J.~ w(t)dt. b. We give a second proof for arbitrary d > 1. Define E = {(t,u) E Rt x Rt: u; < w;(t;), all j}. For u E Ri define a(u) = {t E Rt: (t,u) EE} = {t E Rt: w;(t;) > u;, all j}. Note tha.t, since ea.ch w; is decreasing, each a:(u) is a. (possibly empty) rect- angle in Ri. Therefore, 1 ip(t) dt $ M la:(u)I a(u) 128 for all u ER'.!-. Now, d ITd 1w;(t;) duj IT Wj(ti) - w(t), j=l 0 j=l so - { w(t) dt. JR"+ Since E can also expressed as E { ( t, u) E Rt x R! : t E a( u)}, we have - { dt la(u)I, la.cu> and therefore, { la(u)ldu. }Rd + { 1, let v, w E A(Rd) be such that llv/1 1 = llwll1 = 1 and and As v, w E L1(Rd) we can define u = v * w. Note that llu/'1 = I/viii 1/wlli = 1 . b d smce oth v, w > 0. Also, u(t) = IL (v; *w;)(t;) and v; *Wi E A(R) for all j, sou E A(Rd). Now,/* g exists and is integrable since/, g E AP' c L1(Rd). We compute l(f*g)(t)I - ILdf(t-s)g(s)dsl <(I IJ(t-s)g(s)/P' )l/p'(r )1/p Jad v(t - s)P'/Pw(s)P'/p ds Jad v(t - s)w(s)ds 135 - l"' 1/p' ( lf(t - s)g(s)IP' v'(t - s)w'(s)ds) u(t) 1 IP. Thus, II!* 9ll~P,.', = J(R "' IU * g)(t)lp' u'(t) dt ~ r r lf(t - s)g(s)\P' v'(t - s)w'(s) u(t)P'IPu'(t)dsdt JR4 JR"' = r lg(s)IP' w'(s) r lf(t - s)\P' v'(t - s) dt ds JR"' JRd - (JR"' \g(s)IP' w'(s)ds) (ld lf(t)\P' v'(t)dt) - 119ll~p' 11/ll~P' w 1 11 1 Let ting c -+ 1 therefore gives the result. I 136 Section 3.4. A characterization of B(p, q). In this section we attempt to characterize B(p, q) in a manner similar to Beurling's characterization of B(p, oo) given in Section 3.3, i.e., as a union or intersection of weighted LP spaces. DEFINITION 3.4.1. Given 1 $; p < oo and 1 $ q < oo. a. Given a weight w on R~ we define the weight Wpq on R~ by Wpq(t) - III(t)w(t)l1-~. b. If p $; q then we define X(p, q) = nwEA Ltp/R~), with norm llfllxcp,q) - sup IIJIIL~ wEA, P'l llwlli=l - !~ll_ (fa. I/(t )I' III( t) w( t) \? - ~ \II~:)\) t/p (J.., w( t) dt) H . ? c. If q $; p then we define X(p, q) = UweA Ltn (R~), with norm ll!llxcp,q) inf wEA, llwll1= 1 ( ) 1l~~ JatJ 1 dt /P(Jfa t ).1._.1 ( . lf(t)IP III(t) w(t)l -~ III(t)I w(t) dt " P ? REMARK 3.4.2. a. Since fatJ w(t) dt = fa~ \II(t) w(t) \ in1~)I' the normal- ization of w with respect to Lebesgue measure in Definition 3.4.1 can be considered a normalization of III(t) w(t)I with respect to the Haar measure dt/lII(t)\. 137 b. For q = oo, II /llxc,,=> = !~~ (L: If (t )J? Ill(t ) w( t)I Jn~: )I) ?I? (J... w( t) dt )-'I ? = sup (fa" lf(t)IP w(t) dt) 1/p wEA fa" w( t) dt = ll!IIBP? Thus X (p, oo) = BP = B(p, oo ), cf., Theorem 3.3.9. c. Since Wpp = 1, X(p,p) = V(R~) = B(p,p), cf., Proposition 3.2.2. d. For q = 1, llfllx(p,1) = !.i ~ (J rR : 1 dt ) l/p ( ) 1-? lf(t)IP III(t) w(t)l -p IIT(t)I la1d w(t) dt = !'t,. (la. I~ ~?ii' w(t)'-? dr? u.... w(t ) dt )"'' = llf(t)/II(t)II AP. Thus X(p,1) = Il(t) AP= B(p,1), cf., Corollary 3.3.11. q - p p' - q' LEMMA 3.4.3. If 1 -< p, q < oo then -- = 1 , ? pq pq PROOF: We compute q - p p' - q' 1 1 1 1 - + - - 1 - 1 = o. I pq p'q' p q q' p' LEMMA 3.4.4. Given a weight w on R~ and given 1 :5 p < oo and 1 :5 q .$ oo, 138 P ROOF: By Section 1.7a it suf fi ces to s h ow Wpq 1-p' = w p'q'? Assume first that q < oo; then, with the help 0? Lemma 3.4.3, we compute ( q ~ p) ( 1 - p') = q ~ p 1 ~ p - q-p _P_ pq 1-p I I - p -q (-p') p'q' q' -p' q' Therefore, wpq(t) 1-P' _ III(t)w(t)l<1-p/q)(l-p') - III(t)w(t)l1-p'/q' = Wp 1q1(t). If q = oo then q' = 1, so 1Dpoo(t)1-p' - III(t)w(t)l1-p' - Wp11(t). I REMARK 3.4.5. From Lemma 3.4.4 we can prove that X(p, q)' = X(p', q'), cf., Proposition 3.3.10. Note that B(p, q)' = B(p', q') by Remark 3.2.6c. PROPOSITION 3.4.6. a. lfl :$ p $ q $ oo then X(p,q)::, W.(LP,Lq). b. If 1 < q $ p < oo then X(p,q) C W.(LP,Lq). PROOF: a. For simplicity, assume q < oo as the case q = oo is similar. Fix f E W.(LP,Lq) and w E A(R"). Since (q - p)/q ~ O, w 1), the fact that (q/p)' = q/(q - p), and Lemma 3.3.5c. Therefore, r ) (p-q)/pq llfllLt.,.. (J a,. w(t)dt ~ 22 d(9.-p)/pq llfllw.(L",L?)? Taking the supremum over w E A(Rc l) we obtain llfllx(p,q) :5 22d(q-p)/pq ll!llw.(LP,L?)? b. Fix/ E W.(LP,L") and w E A(Rc1). Since (q - p)/q :5 O, w 5 224CP-q)/pq llfllx(p,q)? ,II 141 REMARK 3.4. 7. a. From Remark 3.4.2, B(p, 1) = X(p, 1), B(p,p) = X(p,p), and B(p, co) = X(p, co). From Proposition 3.4.6, B(p, q) = W..,(V, Lq) C X(p,q) if p < q, and B(p,q) = W..,(V,Lq) ::J X(p,q) if p ~ q. We therefore strongly suspect, although we have not proved, that B(p, q) = X(p, q) for all P, q. b. In the deep paper [He], Herz introduced spaces related to X(p, q). Using the notation of [Jo), the Herz space pLq is defined as follows. Let cl> be the spherical analogue of A(Rd), i.e., cl> consists of all weights w on Rd which are positive, radial, integrable with respect to Lebesgue measure, and radially decreasing. Then 1/p(L ) .!._.7!, sup (l a/ ., lf(t)IP ,p(t) 1-~ dt ) ,p(t) dt v , p :5 q, cpE+ B." inf ( / IJ(t)IP ,p(t)1-~ dt) l/p ( f ,p(t) dt) ?-?, ~? ~ k, p~ q. The space K!q is defined by the norm llfllK0 = II jtjC;-?)d f(t)jj L ? JJ'l JJ 'l Since K!q is the analogue of X(p, q) obtained by using Lebesgue measure on Rd instead of Haar measure on R~, and using a spherical approach to higher dimensions rather than a rectangular approach. 142 The space K~ is defined by the norm lllllx:. = II 1w?1" f(t>llx: ?. Therefore Kp-qP = ltl-p/d K 0 pq is the exact spherical analogue of the rectangular X(p,q). 143 Section 3.5. Weighted Besicovitch spaces. In this section, we examine the effect of replacing the factor l/lRTI in the definition of II? IIB(p,q) by a general function p(T). We show that the resulting space, denoted Bp(p,q), equals a Wiener amalgam space W.(Vv,L9 ) for an appropriate weight v. The spaces Bp(p, q), especially Bp(p, oo ), have appeared in various places in the literature. For example, Wiener considered the one-dimensionsal case 1/(2T)'\ e.g., [W3), as did Lau and Chen, e.g., [CLI]. Strichartz consid- ered higher-dimensional spherical analogues of this, e.g., [Stl; St2]. Evans considered general functions, e.g., [E2]. DEFINITION 3.5.1. Given 1 ~ p, q < oo and a weight p: Ri -+ R+, the weighted Besicovitch space B p(P, q) is the space of functions /:Rd -+ C for which the norm J.fa ." ( f ) q/p dT ) i/q llfllBp(p,q) = ( p(T) }RT lf(t)IP dt II(T) + is finite. The standard adjustments are made if p or q is infinity, cf., Definition 3.2.1. The following result is similar to part of Theorem 3.2.4. PROPOSITION 3.5.2. Given 1 $ p < oo, 1 $ q $ oo, and an even, moderate weight v on R:. Define p(t) - v(t)/III(t)j. 144 Then there is a. constant C > 0 such that II ? llsp(p,q) > C 11 ? llw.(Lt,L?>? PROOF: Assume for simplicity that 1 ~ p, q < oo (the case q = oo is sim- ilar). By Theorem 2.1.6e there is a constant B > 0 such that SUPt[l,2] v ~ B inft[1,2J v for t E R~. Therefore, if T E [2", 2n+i] then and v(T) > v(2") p(2") p(T) = II(T) B II(2n+i) - 2d B . Fix now any (T E nd. Then 111111.,(.,.) = Lt (p(T) L. lf(t)I' dtt? II~) = L I (p(T) I 1/(t)IP dt)q/p IId(TT) n J[2?,2,.+1] }Rx > (log 2)d L (p~") 1 1/(t)IP dt) q/p n 2 B a-[2n-1,2aJ > (log2)d L (22lB21 1/(t)IP III(t)lp(t) IIId(tt)l)q/p, n a-[2?- 1 ,2"] 145 where the summations in n run over zd. Therefore, > (log2)d 2-2dq/p B-2q/p L (1 IJ(t)IP v(t) __!:!_)q/p n,o- o-[2?- 1 ,2?) III(t)I (log 2)d 2-2dq/p B-2q/p llillq ,. W.(Lv,L?)' from which the result follows. I REMARK 3.5.3. a. The opposite inequality to the one in Proposition 3.5.2 can be proven just as in Theorem 3.2.4. Precisely, given 1 < p < oo and 1 ~ q ~ 00 and given an even moderate weight p: R~ ~ R+, define v(t) = III(t)I p(t). Then there exists a constant D > 0 such that II? IIBp(p,q) ~ D 11 ? llw.(Lt,L?)? b. From Proposition 3.5.2 and part a we have Bp(p,q) = W.(Lf,Lq) with equivalent norms. c. By Theorem 2.3.1, if vis moderate then w.(L!,Lq) = w.(LP,L:.,,,,), i.e., the weight may be placed on either the local or global component. 146 CHAPTER 4 THE WIENER TRANSFORM In this chapter, we prove that the Wiener transform W is a topological isomorphism of the Besicovitch space B{2, q) onto the variation space V{2, q) for each 1 :::; q :::; oo. The definition of the Wiener transform and the symmetric difference oper- ator -6..x used in this thesis follow the higher-dimensional, rectangular defini- tions of [BBE]. Many basic ideas in this chapter are from [BBE]; we thank those authors for making higher-dimensional calculations possible. In addi- tion, the one-dimensional results of Beurling and Lau are critical in that they directly lead to our isomorphism theorem. Our method of proof includes new techniques based on amalgams, combined with the techniques of Beurling and Lau. We begin in Section 4.1 by defining the Wiener transform, and showing that its domain of definition includes the spaces B{2, q) for each 1 :::; q :::; oo. In Section 4.2 we define the symmetric difference operators -6..x, and com- pute -6..x W f. In Section 4.3 we define the higher-dimensional variation spaces V(p, q). In Section 4.4 we prove that the Wiener transform maps B{2, q) continu- ously into V{2, q). We prove this for the case q = oo using Lau's method, for q = l using Beurling's method, and for the general case using amalgam 147 spaces. In Section 4.5 we prove that the Wiener transform is invertible for each 1 ~ q ~ oo. We prove this for 1 ~ q ~ oo by generalizing Lau's q = oo technique, and compare this to an amalgam space proof for 1 ~ q < oo. Throughout this chapter, k and K will be as in Example 3.3.4c. That is, 2 k(t) = ((sin21rt)/(1rt)) fort ER and 2 d K(t) = rr(sin21rt) = II(sin21rtj)2 II k(t;) 1rt 1rt,? j=l j=l k * denotes the least decreasing majorant of k on R+, extended evenly to R, and k* the greatest decreasing minorant on R+, extended evenly to R. k* is even, positive, integrable, and decreasing on R+, and therefore is an element of A(R). k* is even, nonnegative, integrable, and decreasing on R+, but is not an element of A(R) as it has zeroes. In fact, k* = k-X[o,1; 21. Numerically, 100 00 00 k*(t) dt ~ 1.068, 1 k(t) dt = 1, and 1 k*(t) dt ~ 0.903. Similarly, K* denotes the least decreasing majorant of K on Ri, extended evenly to Rd, and K* the greatest decreasing minorant on Ri, extended evenly to Rd. K* is rectangular, even, positive, integrable, and decreasing on Ri, and therefore is an element of A(Rd). K* is rectangular, even, non- negative, integrable, and decreasing on Ri, but is not an element of A(Rd). 148 Section 4.1. Definitions. In this section we define the Wiener transform and show that its domain of definition includes each Besicovitch space B(2, q) for 1 < q ~ oo. DEFINITION 4.1.1. Given t E Rd and -y E Rd we define d -211'it?""? X (t ) t:(t,-y) - II e ' '' - (-1 ,1] j -21rit; j=l Note that if lt;I > 1 for all j then ?(t,-y) = E_7 (t)/II(-21rit). DEFINITION 4.1.2. Given a function /:Rd--+ C, its Wiener transform is (formally) Wf('Y) JRI. ,_ f(t) t:(t,-y) dt, Wiener denoted Wf bys, a notation retained in [BBE], where it is called the Wieners-function. The integral defining the Wiener transform may converge in various senses, depending on the function/. For example, it may converge absolutely or only in mean, cf., Example 4.1.4 and the proof of Theorem 4.1.7. LEMMA 4.1.3. Given (t,-y) E Rd x Rd. a. III(t) ?(t,-y)I ~ 11'-d. b. 1?(t,-y)I :S (21r)la(t)l-d IJ;Ea(t) 1-Y;I, where a(t) = {j: It;/~ 1}. c. SUPtE(-1,1] /t:(t,-y)/ '.S /II('Y)l- 149 PROOF: a. We compute -d 7r b. Fix t E Rd and, E ftd. If t; E (-1, 1] then e-2,rit;-r; - 11 < 121rt;,; I = I - 2 7r1,? t j - 2 1rt; l,;I- On the other hand, if t; (!. [-1, 1] then The result therefore follows by multiplication. c. Follows immediately from b since t E [-1, 1] implies a(t) = {1, ... , d}. I We give examples of the various senses in which the Wiener transform may converge. EXAMPLE 4.1.4. a. Given f E L1(R!), we have from Lemma 4.l.3a that Ld lf(t) t'(t,,)I dt $ 7f'-d L~ lf(t)I III~:)I = 7f'-d II/IIL1 (R~) < 00. Thus W f converges absolutely and IIW /11 00 $ 1r-d 11/11 1 , so Wis a continuous Note that since B(p, 1) C LP(R!)nL1 (R!), the Wiener transform converges absolutely for functions in B(p, 1 ). b. Set d = 1 and fix f E L00 (R). Since 1 2 1 1 1 jJ(t) e- ,rit-y __ 1 dt $ hi [ lf(t)I dt < 21,111/lloo, _ 1 -2nt 1 150 the integral 1 e-21rit-y _ 1 1 (4.1.1) -1 f(t) -21rit dt converges absolutely. Now define g(t) J(t) X (t) 2 ?t (-00,-l]U[l,00) ? - ,ri Then 100 lu(t)l2 dt ~ llfll~ /00 r2 dt < oo, -00 2,r 11 so g E L 2 (R). Therefore, its Fourier transform g converges in mean. Evalu- ating, e-21rit-y _ = 1 1 ( 4.1.2) g(-y) j(t) . dt. itl>l -2,rit The Wiener transform off is the sum of the two integrals ( 4.1.1) and ( 4.1.2), so is well-defined. Moreover, Wf E Li:c(:R) + L2(R) C Lf0 c(R). We partition Rd and zd into subsets R~ and Z~ as follows. DEFINITION 4.1.5. Given a subset a C {l, ... , d}. a. R~ = {t E Rd : It;! < 1 for j Ea, lt;I > 1 for j ft a}. b. Z ~ = {n E zd : n; < 0 for j E a, n; ;:::: 0 for j ft a}. REMARK 4.1.6. a. {R~} is a partition of Rd and {Z~} is a partition of zd. b. If a = {1, ... , d} then R~ = (-1, 1). All other R~ are unbounded and disconnected, consisting of 2d-Jal connected components. 151 THEOREM 4.1. 7. The Wiener transform is de-fined on W.(L 2 , L00 ) and is a continuous linear map of W.(L 2 , L00 ) into LfaJftd). PROOF: Fix f E W.(?2 , ? 00 ). It suffices to show that Fa(,) = f J(t)?(t,,)dt la"a is well-defined and an element of Lf cCR.d) for each a C {1, ... , d}, and that 0 the mapping / i--+ Fa is continuous. Recall that 1 2 2 dt ) / llfllw.(L2,L 00 ) = sup (1 lf(t)l III(t)I nEZ", o-EO" o-(2" ,2"+ 1] a. Assume first that a = {1, ... , d}, and note the following facts. a2. t ER~ => l?(t,,)I ~ /II(,)/. a3. n E Z~ => ni < 0 for all j. a4. Z::neZ" II(2") = 1. a In the following calculation, the summations are over n E Z~ and e(t,,) = E_.y(t)/II(-21rit). b3. n E Z# => n; ~ 0 for all j. In the following calculation, the summations are over n E z: and . ER, ~ d Therefore, for >. E R , A 1+ Ad+ ~~l ? ? ? ~~d f. Assume F is rectangular in the sense of Section 1.3d, 1.e., F(,) 160 il).F(,) d II Ll{j F;(,j) j=l d 2-d II [F;(,j + Aj) - F;(,j - Aj)]. j=l g. Since il). is a sum of translation operators, it is a tempered distribution. Therefore, its Fourier transform exists and is a sum of modulation operators. I\ We denote this operator by il).. Since the act of modulation is multiplication I\ by an exponential, il). acts by multiplying by a function which is a sum of I\ exponentials. We denote this function by il)., i.e., V Similarly, il). is the function which is the inverse Fourier transform of b. ).. We evaluate these functions in Proposition 4.2.5. DEFINITION 4.2.3. Given A E ftd. a. The sine product function S). is d II( sin 21r At) II sin27rAjtj. j=l b. The Dirichlet kernel d). is Id Isin-21rA-jtj j=l 1rt3? REMARK 4.2.4. a. JJs).lloo = 1. 161 c. d>. E V(Rd) for 1 < p ~ oo, and d>. ~ L 1 (Rd). d. K(t) = ld1(t)l 2 and ld>.(t)l 2 = Il(~)2 K(~t). A V PROPOSITION 4.2.5 . .6.>.(t) = ids>.(t) and .6.>.(t) = (-i)ds>.(t). PROOF: We prove only the first statement as the second is similar. We com- pute 1>.(t) - (2-d L IT(u)T-u>.)A(t) uenc1 2-d L IT(u)Eu>.(t) venc1 v1E{-l,l} vc1E{-l,l} d - 2-d II j=l u;E{-1 11} d - 2-d II (e2'ri>.;t; -e-21ri>.;t;) j=l d _ 2-d II 2i sin 271" ~;t; j=l We characterize in the following proposition those functions F such that .6.>,.F(,) = 0 for all ~- For example, constant functions satisfy this condition. In one dimension there are no other examples. PROPOSITION 4.2.6. a. A function F: Rd-+ C with (4.2.1) is completely determined by its values on the coordinate hyperplanes Ad = 162 { "'Y E Rd : II("'Y) = 0}. Conversely, every function F: Ad --+ C uniquely determines a function F on Rd which satisfies (4.2.1). b. Ford= 1, a function F satisfies (4.2.1) if and only if it is constant. c. Ford> 1, all constant functions satisfy (4.2.1), but they do not exhaust the class of F satisfying (4.2.1). PROOF: Assume first that d = 1, and recall that A~F("'Y) = ?[ F("'Y + A) - F("'Y - A)]. Setting "'Y = A, we find 0 = 2A-yF("'Y) = F(2"'Y) - F(0). Thus F("'Y) = F(0) for all "'Y, so Fis constant. Now assume that d = 2 ( the general case is similar). Recall that A~F("'Y) = ?[ F("'Y1 + A1,"'Y2 + A2) - F("'Y1 + A1,"'Y2 - A2) - F("'Y1 - A1,"'Y2 + A2) + F("'Y1 - A1,"'Y2 - A2)]. Setting "'Y = A, we find The last three terms of this expression lie on the coordinate axes, so the value of F(2"'Y1, 2...,,2) is completely determined by the values of F on the coordinate axes. Conversely, assume F: A 2 --+ C is given. Extend F to R 2 by defining F("'Y1,"'Y2) = F("'Y1, 0) + F(0,""12) - F(0, 0). Given any "'Y, A E R.2 , we then have 4A~F("'Y) = F("'Y1 + A1,"'Y2 + A2) - F("'Y1 + A1,"'Y2 - A2) F("'Y1 - A1, "'Y2 + A2) + F("'Y1 - A1, "'Y2 - A2) 163 - F(0,,2+.X2) + F(0,0) + F(0, ,2 - .X2) - F(0, 0) + F(0,,2+.X2) - F(o,o) - F(0,,2 - .X2) + F(0,0) - 0. Thus F satisfies (4.2.1). I We apply the difference operator to t:(t, ?). LEMMA 4.2.7 [BBE]. ~.\t:(t, 1 ) = 2-d E_-y(t)d.\(t). PROOF: By Remark 4.2.2d, it suffices to show ~{t:(t,,) = ?E -'>'; (t;) d.\; (t; ). Since this calculation is the same as the one-dimensional case, we assume d = 1 and compute 2~.\?(t,,) - ?(t,,+.X) - t:(t,,-.X) e-21rlt{'>'+.\) - X[-1,1](t) - e-21rlt('>'-.\) + X[-1,1J(t) -21rit 164 = e-21rit-y sin21rAt 1rt PROPOSITION 4.2.8. Given 1 ~ q ~ oo, f E B(2,q), and A E Ri, a. f ? d>,. E L 2 (Rd). b. A>,.Wj = 2-d (f ? d>,.)" E L 2 (Rd). PROOF: a. Since B(2, q) C B(2, oo) = W.(L 2 00, L ), it suffices to prove the result for f E W.(L 2 ,L00 ). Note that (4.2.2) Since K*(At) E A(Rd) for each fixed A, we have by Lemma 3.3.Sc that From (4.2.2), (4.2.3), and the fact that K* is even and decreasing on Ri, we therefore have 165 :::; 23d IT(.\) II/II~. (L2,L"") laI+" K*(t) dt < 00. b. Fix/ E B(2,q). From Theorem 4.1.7, Wf is well-defined and is an element of Lf c(:Rd). Using part a and Lemma 4.2.7, we compute 0 A.\Wf(-y) A.\ I f(t) t:(t,-y) dt la" I f(t)A.\t:(t,-y)dt la" The fact that A.\ can be interchanged with the integral in the above cal- culation follows immediately from the fact that A.\ is a sum of translation operators acting only on -y. I REMARK 4.2.9. In [BBE], Proposition 4.2.8 is proved (using different esti- mates) for all/ E L;(R) :) B(2,q), where vis as in Theorem 3.3.1 with a= 2. 166 Section 4.3. The variation spaces. In this section we define the variation spaces V(p, q). Definitions for d = 1 were given in (0.2.16). DEFINITION 4.3.1. Given 1 ~ p, q < oo, the variation space V(p,q) is the space of functions F: ftd--+ C for which the seminorm is finite. The standard adjustments are made if p or q is infinity, namely, IIFl/v(p,oo) IIFllv(oo,q) IIFllv(oo,oo) We also define V(p,lim) = {F: 2lim rr (d-\ ) J/a , l~-\F("Y)IP d"'f exists}, -\--+o where the limit is the d-dimensional limit defined in Section 3.1. REMARK 4.3.2. 11 ? llv(p,q) is not a norm, since IIFllv(p,q) = 0 implies only that ~-\F(-y) = 0 for a.e. "Y and A. For example, all constant functions F satisfy IIFllv(p,q) = 0, cf., Proposition 4.2.6. However, II ? llv(p,q) is a seminorm, and therefore becomes a norm once we identify functions F, G E V (p, q) such that IIF - Gllv(p,q) = 0. We adopt this convention for the remainder of this 167 - -"'::".:'' _.,_.,.._., ..;: --? ?....,dS thesis, so V(p, q) is at least a normed linear space. We prove in Theorem 5,2.3 that V(p, q) is complete, hence a Banach space. The proof of this fact is complicated by the fact that V(p, q) is not solid, i.e., given F, GE V(p, q) with IFI < IGI a.e., we cannot conclude that 1/Fl/v(p,q) $ /IG/lv(p,q), cf., Example 4.3.3. We will not need the completeness of V(p, q) for any results in this chapter. EXAMPLE 4.3.3. Set d = 1, q = oo, F = X[o,1J, and G = 1. Then we have /IGllvcp,) = O, while IIFllv(p,oo) > 0 since d = 1 and F is not identically constant (Proposition 4.2.6). In fact, since "'I .~,1, ,,: u. ' j I ' We have ,.~, _ { 4A, O 0 = IIGllv(p,oo)? REMARK 4.3.4. a. The Wiener-Plancherel formula, as proved by Benedetto, Benke, and Evans in higher-dimensions, states that the Wiener transform W is an isometry of the nonlinear space B(2, lim) onto V(2, Iim). b. Lau and Chen proved that, ford== 1, Wis a topological isomorphism of B(2, 00) onto V(2, 00 ). We discuss this result in Sections 4.4-4.5. 168 c. Beurling proved that, for d = l, the Fourier transform is a topological isomorphism of A2 onto V(2, 1). We show in Sections 4.4-4.5 that this implies that the Wiener transform W is a topological isomorphism of B(2, 1) onto V(2, 1). d. We prove in Sections 4.4-4.5, for arbitrary d ~ 1, that Wis a topological isomorphism of B(2, q) onto V(2, q), for ea.ch 1 ~ q ~ oo. EXAMPLE 4.3.5. Assume f is given and its Wiener transform W f is well- defined, e.g., f E B(2,q). Note that ld.\(t)l 2 = II(>.)2 K(>.t). Therefore, by Proposition 4.2.8 and the Plancherel theorem, IIWfllv<2,,J - (fa/rr~:) k_, ld,Wf (-r)I' d-y /' rr~1) f' (( (II ~:) k_, 1r' (f. d,)'(-r)I' d-y) q/2 rr~1)) 1/q / ( 1 f )q/2 d>. )1/q ( lad 2d II(>.) JRtl lf(t) d.\(t)l2 dt IT(>.) + ( r (rr(>.) r )q/2 d). )1/q l"fl.tl 2?l JRtl lf(t)l2 K(>.t) dt IT(>.) . + 169 Section 4.4. Continuity of the Wiener transform. In this section we prove that the Wiener transform is a continuous mapping of B(2, q) into V(2, q) for each 1 ::; q ::; oo. We begin by examining Lau's proof for the case q = oo. Next, we show that Beurling's proof that the Fourier transform is a continuous map of A2 into V(2, I) implies that the Wiener transform is a continuous linear map of B(2, 1) into V(2, I). Finally, we prove the general case by using amalgam space techniques. The following proposition, for the cased= 1, is due to Lau and Chen. PROPOSITION 4.4.1. Given a rectangular, positive, even function w, and given 1 ::; p < oo. Then sup II(;) lafd IJ(t)IP w(.Xt) dt < llfll~(p,oo) ~ead+ 2 la / d w*(t) dt + for all measurable functions f: Rd-+ C. PROOF: Assume without loss of generality that fad w*(t) dt < oo and that + f E B(p, oo ). Extend w * evenly to Rd and note that (4.4.1) IT(;) f lf(t)IP w(.Xt) dt = 2-d f lf(t/ ;\)IP w(t) dt 2 lad lad for all A E Ri, where D~ is the usual dilation operator. Recall from Theorem 3.3.9 that B(p, 00) =BP= nweA L~(Rd), with norm equality, i.e., fad lg(t)IP w(t) dt - !~~ fad w(t) dt ? 170 Since B(p, oo) is dilation isometric and w* E A(Rd ), we therefore have ( 4.4.2) Jra,J JD>.f(t)JP w*(t) dt ::; JID>./lltc =) P, lar,J w*(t) dt The result follows upon combining (4.4.1) and (4.4.2). I REMARK 4.4.2. Lau and Chen prove in [CLl] that fa" w*(t) dt is the best + possible constant in Proposition 4.4.1. Their proof of this fact is intricate, and will be omitted. We point out, however, that it carries over immediately to higher dimensions. COROLLARY 4.4.3. The Wiener transform Wis a continuous linear map of B(2, oo) into V(2, oo ), with IIWII = (/.~ k*(t)dt)''' "' (1.033)4 > 1. PROOF: From Example 4.3.5, IIW fllh2,=) = sup II(;) r lf(t)l2 K(.Xt) dt . .\ER,J 2 Ja,J + The result therefore follows from Proposition 4.4.1, Remark 4.4.2, and the fact that fatJ K*(t)dt = (fo00 k*(t)dt)d. I + We turn now to Beurling's proof that the Fourier transform is a continuous linear mapping of A2 into V(2, 1), which we recast as showing that the Wiener transform maps B(2, 1) continuously into V(2, 1). The critical fact, and our starting point, is the following nontrivial result, also due to Beurling, e.g., [Be2]. 171 LEMMA 4.4.4. Given w E A(R) and given O < a < 1 < b < oo, there exists a function w? such that a. w* ~ w, b. ta w*(t) is decreasing on R+, c. tb w*(t) is increasing on R+, J00 00 d. 0 w?(t) dt ~ (l-a)(b-l) f0 w(t) dt. PROPOSITION 4.4.5. The Wiener transform Wis a continuous linear map of B(2, 1) into V(2, 1), with PROOF: Fix f E B(2, 1); then J(t)/IT(t) E A2 by Corollary 3.3.11, and llf llB(2,l) - 2-d IIJ(t)/Il(t)IIA2 2-d inf ( f IJ(t)/II(t)I 2d t )1/2 (L w(t) dt )1/2 wEA JR" w(t) R" Fix any w E A(Rd) with llwll 1 = JR.t w(t) dt = 1. By definition, w(t) = Ilt w;(t;) for some w; E A(R). Let wj be the functions given by Lemma 4.4.4 applied to w; with a = 1/2 and b = 3/2. Define w?(t) = IIt wj(t; ). Then d foo ( 4.4.3) f w*(t) dt }Rtt IT Jo w;(t;) dt; + j=l 0 ]dl 3/2 loo < (1/2)(1/2) Jo w;(t;)dt; 172 = 6d f w(t)dt 1a4 + Now, by Proposition 4.2.8 and the Plancherel theorem, (4.4.4) 77(.X) - II~:) l IA.\ W fh')l 2 d,y 4 2 2d ~(.X) L4 lf(t) d.\(t)l dt - (21r2 )! II(.X) L 2 js.\(t) f(t)/IT(t)l dt. 4 Using ( 4.4.3) and ( 4.4.4) we therefore compute (4.4.5) J(a . 1/2 d.X IIW fllv(2,1) = 41 1(.X) TI(.X) + = f ( 11P) ) 1/2 (w*(l/21r.X)) 1/2 d.X Ja.4 w?(t/21r.X) II(.X)2 + Now, t/l2wj(t;) is decreasing on R+ and t/l2wj(t;) is increasing on R+ for each j. Therefore, given t;, /3; E R+, 0 < /3; ~ 1 => t/12 w;(t;) < (t;//3;)312 w;(t;//3;), 1 ~ /3; => t/12 w;(t;) < (t;//3;) 112 wJ(t;//3;), 173 whence 1 ~ /3j ? wJ(tj//3j) 2'. f3/12w J(tJ). Therefore, ( 4.4.6) 00 / /sA; (ti)/ 2 d>.i lo wj(1/21r>.j) >.i 00 - 1 sin 2 21r>. ?t ? d>. ? ) ) ) - 0 wj(I/21r>.i) >.i 00 2 / sin /Ji d/3i = lo wj(ti//3i) /Ji < /1 2 sin /Ji d/3i + 100 2 sin /Ji d/3i - lo f3/12w j(ti) /Ji f3/ 12 1 wj(ti) /Ji 1 [1 / 00 sin/Ji /2 d/3i 1 1 ? 2 d/3j = wj(tJ) lo T; f3/12 + wj(ti) 1 sm /Ji /3//2 < -1- 11 da . _fJ)_ + -1- loo d/3)? - Wi(ti) 0 /3//2 w;(ti) 1 /3//2 4 wi(ti)' from which it follows that ( 4.4. 7) Substituting ( 4.4. 7) into ( 4.4.5), (12) d/2 (1 lf(t)/II(t)/2 ) 1/2 //Wfl/v(21) ~ - () dt . ' 7r R" W t Since this is true for all w E A(Rd) with //w/1 1 = 1, 2 12 12 1/W fllv(2,1) ~ ( ~ )' 1/J(t)/II(t)I/A? = (: )' 1/flls(2,1)? I 174 REMARK 4.4.6. a. If we repeat the calculations in the proof of Proposition 4.4.5, keeping all estimates the same but not fixing a and b, we find that 2 < ( 2b(2+a-b) )d/ IIW/llv(2,1) _ 1ra(b - l)(l _ a)(2 _ b) llflls(2,1)? The expression F b _ 2b (2 + a - b) (a,) - 1ra(b-1)(1-a)(2-b) is clearly is not minimized at a,= 1/2, b = 3/2, but at 2 bo _1 (9_ _ ---4--=?-B--1 _ ?( --9 + ---4--=}-B--1 ) - 2B2 - 2B ) 1 2 2 B 2 2 B2 ' where 4 208 (2971 ?-6373)-l/3 113 (2971 ?-6373) 3 + 9 27 + 3J3 + 27 + 3J3 2 ?-2+?A1 2 , 3 3 14 + 160 (- 1846 2J-6373)-l/ + (- 1846 + 2J-6373) l/ 3 9 27 + 3J3 27 3J3 This follows from solving for the critical points of F, using that -4b + 8ab + 2a2 b + 2b2 - 4ab2 1ra2 (a -1)2 (2 - b)(b -1) ' 8 + 4a - 8b + 2b2 - 2ab2 1ra(a -1) (b - 2)2 (b -1)2 ? 175 A 1 , A2, B 1 , and B 2 are real. Numerically, a0 ~ 0.352 and bo ~ 1.528, and F( ao, b0 ) ~ 43.904/71" < 48/71" F(l/2, 3/2). Therefore, IIWJ!lvc2,1) ~ (3.738)d ll!IIB(2,1)? b. If we repeat the calculations in the proof of Proposition 4.4.5 but without fixing a and band without approximating (sint)/t and sint in (4.4.6), we find that IIWfllvc2,1) < G(a,b)d12 IIJIIB(2,1), where G(a,b) - Using numerical integration, we compute G(l/2, 3/2) ~ 36.85/7r and G(ao,bo) ~ 32.30/71", both of which improve on the estimates in part a. G is minimized at a1 ~ 0.30, b1 ~ 1.54, with G( a1 , b1 ) ~ 31.92/71". Thus, IIWJ!lv(2,1) < (3.19)d IIJIIB(2,1)? We do not know if this is the best possible constant. Beurling's and Lau's results establish the "endpoints" of our isomorphism theorem. The "midpoint" is proved in the following proposition. 176 PROPOSITION 4.4. 7. The Wiener transform is an isometry of B(2, 2) into V(2, 2). PROOF: We compute, with the help of Example 4.3.5 and Proposition 3.2.2, IIW//lh2,2) - I IT(A) JRI ,,. lf(t)l 2 K(>.t) dt ~ ls." 2d IT(>.) + = 2-d JRf ,,. lf(t)l 2 f K(>.t) d). dt J'fLd. + = 2-d L,,. lf(t)l2 fa,,. K(>.) d>. 1n~!)1 + -d I 2 dt = 2 JR,,. lf(t)I IIT(t)I - ll/111c2,2)? 1 REMARK 4.4.8. W is surjective by Theorem 4.5.5, so is actually a unitary map of B(2, 2) onto V(2, 2). As II? llnc2,2) = 2-d/2/ I? llv(R~), Wis therefore a multiple of a unitary map of L2 (R~) onto V(2, 2). Next we prove the continuity of the Wiener transform on B(2,q) for 2 :s; q ~ oo by using amalgam space methods. The constants we obtain are not best possible, cf., Remark 4.4.16. PROPOSITION 4.4.9. Given 1 :s; p ~ q ~ oo with p f:. oo, and given a nonnegative, even function w on Rd. Then 177 (l.(rr~;) L, lf(t)I' w(>.t)dt)'1' rr~~)r q + '.S (log 2)dfq 2??!? (L, w*(t) dt) l/p llfllw.(L?,L?) + for all measurable functions f: Rd -----+ C (with the standard adjustments if q = oo). PROOF: Extend w* evenly to Rd and assume q < oo (the case q = oo is similar). The summations in the following calculation are over m, n E zd and u E _nd_ Using the fact that w* is even and decreasing on Ri, we compute ( 4.4.8) la/ d (II(>.) / )q/p d>. ~ }Rd lf(t)IP w(>.t) dt IT(>.) + < L I (IIC:~+1) L r IJ(t)IPw*(2m+n)dt)q/p ~ n J[2n,2n+l] :Z... m,17 J.,.[2m,2.,,,.+1] II().) (log 2)d L (L II(2") w*(2m+n) / lf(t)IP dt) q/p n m,17 } 17[2m ,2m+l] n m,17 (log 2)d 2dq/p II L Fm,.,. 11:::P, m,17 178 where Frn,a- is the sequence Since q/p "2: 1, we can apply Minkowski's inequality in the Banach space fq/p to estimate III: Frn,a-lltq/p, i.e., ( 4.4.9) IIL Fm,a-lL/p rn,a- < ~ II(2m)w*(2m) (~ (1[,---,,--?+'] lf(t)l' 1rr~!i1) '''t. 2d ~ II(2m)w*{2m) (~ (li,.,,..,] lf(t)J' JII~:)J) ?i-r/? 2 < 2 d (L" w*(t)dt) 11111~.(LP,Lq)' + the last line following from Lemma 3.3.5c. The result follows upon combining (4.4.8) and (4.4.9). I COROLLARY 4.4.10. Given 2 ~ q ~ oo, the Wiener transform Wis a contin- uous linear map of B(2, q) into V(2, q), with 179 PROOF: From Example 4.3.5 and Proposition 4.4.9, ( 4.4.10) IIWfllv(2,q) (l. (II~;) f,,_, lf(t)I' K(>.t)dt) ?I' II~~)) ?I? + (!,,_, 112 < (log 2)dfq 23 2 d/ K*( t) dt) llfllw.(L',L?)? + Now, ( 4.4.11) laI d K*(t) dt (t k*(t)dt) d, + and, by Theorem 3.2.4, ( 4.4.12) The result follows upon combining ( 4.4.10), ( 4.4.11), and ( 4.4.12). I REMARK 4.4.11. For q = 2 and q = oo we know the actual value of JJWII, which we can compare to the estimate for IIWII given by Corollary 4.4.10. For q = 2, JJWII = 1 by Proposition 4.4.7, while Corollary 4.4.10 implies 2 only that !IWII _s; (27 fa?? k*(t) dt)d/ ~ (11.69)d. For q = CX), II WII = (fa?? k *( t) dt) 2 d/ ~ (1.03)d by Corollary 4.4.3, while 00 2 Corollary 4.4.10 implies only that jjWjj :::; (26 f k*(t) dt) d/ ~ (8.27)d. 0 Finally, we prove the continuity of the Wiener transform on B(2, q) for 1 < q :S: 2 using amalgam space methods. PROPOSITION 4.4.12. Given 1 :::; q :S: p < CX), and given a nonnegative, even function w on Rd. Then 180 :S (log 2)"/? 2?/? ( I; (II(2m) w*(2m))'1') l/q 11/llw.(L?,L?) mEZd for all measurable functions f: Rd ---+ C. PROOF: Extend w* evenly to Rd. Just as in (4.4.8), we have ( 4.4.13) fa, (rr~;) !,)f(t)l'w(>.t)dr? II~~) < (log 2)d 2dq/p IIL Fm,u 11::;J>, + m,u ,. j'. where Fm,u is the sequence .. 'l .~.: : .',."1 and m, n range over zd while u ranges over nd. Since O < q/p ~ 1, we can ap- <.f ply the triangle inequality in the metric space p_q/p to estimate III: Fm,u lltg/p, 1.e., ( 4.4.14) III: Fm,u11::;J> m,u ffi 10' m,u n ~ ~ ( rr(2m)w*(2m) f.1,--?,,--?+'11/(tll' 1rr~!i1) ?I? L (II(2m) w*(2m)t/p L (1 lf(t)jP _.!!:!_) q/p m n,u u[2",2"+1] jII(t)j llfll~.(LJ>,Lq) L (II(2m) w*(2m)t/p. m The result follows upon combining ( 4.4.13) and ( 4.4.14). I 181 LEMMA 4.4.13. Given a nonnegative, decreasing function won R+ and given 0 < p < 1, 00 I: 2(2" w(2")Y < P ( sup w(t)P + f w(t)P dt). nEZ 2P -1 099 111 PROOF: Set M = sup099 w(t)P. Then L 2P (2" w(2")Y ~ M L 2"P M-- 2P - 1 n::;o If n > 0 then 2"P ~ 2". Since w is decreasing, we therefore have I: (2"w(2")Y n>O 2P L 12np < -- w(t)Pdt 2P - 1 O 2(n-l)p n> 2 00 P 11f w(t)Pdt. I 2P -1 1 j I =,, LEMMA 4.4.14. Given 1/2 < p ~ 1, ;,.J . ', ,? .. < oo. PROOF: First note that since p > 1/2, 1 2 < 00. 71" P (1 - 2p) Also, sup k*(t)P k*(o)P 099 Since L (II{2m) K"(2m))P = ( L (2" k*(2"))"r mEZd nEZ the result follows from Lemma 4.4.13. I 182 COROLLARY 4.4.15. Given 1 < q::; 2, the Wiener transform Wis a contin- uous linear map of B(2, q) into V(2, q), with PROOF: From Example 4.3.5 and Proposition 4.4.12, ( 4.4.15) U:e., C1IIW fllv(2,,) !;i L IJ(t)I' K(>.t) di)"' II~~) r + < (log 2)'1? 2?/2( I: (II(2m) K*(2m)) ,;, r/? llfll w.(L',L?)? mEZ" From Corollary 4.4.14, 2 ( 4.4.16) L (n(2m) K*(2m)) q/ < ( 2q/2 (2q + 1t1' ? k*(t)q/2 dt)) d 2q/2 - 1 mEZ" And, by Theorem 3.2.4, (4.4.17) The result follows upon combining ( 4.4.15), ( 4.4.16), and ( 4.4.17). I REMARK 4.4.16. a. The combination of Corollary 4.4.3, Proposition 4.4.5, Corollary 4.4.10, and Corollary 4.4.15 establish that the Wiener transform W is a continuous linear mapping of B(2, q) into V(2, q) for each 1 ::; q ::; oo. In summary, we used techniques due to Beurling for the case q = 1, techniques due to Lau for q = oo, and amalgam space techniques for 1 < q ::; oo. Our amalgam space estimate for jjWIJ goes to infinity as q -+ 1, and is inferior to Lau's exact estimate at the other endpoint, q = oo. It is undoubtably 183 possible to improve the estimates for W which we derived using amalgam space methods. For 1 < q < 2, it is likely that there is an amalgam space proof which does not exhibit the "blowing up" effect of the norm as q ---+ 1. b. We do not believe that either Beurling's or Lau's methods can be adapted to prove that the Wiener transform is continuous when 1 < q < oo. In the next section, we prove the W is invertible and derive estimates for II w-1 11 for each 1 _:::; q _:::; oo. Again, Beurling's methods suffice for q = 1, Lau's for q = oo, and amalgam spaces for 1 :::; q < oo. However, Lau's method generalizes easily to all 1 _:::; q _:::; oo. I I l :,.?i. . , 184 Section 4.5. lnvertibility of the Wiener transform. In the preceding section, we proved that the Wiener transform W is a continuous linear map of B(2, q) into V(2, q) for each 1 S q S oo. We proved this for q = l using a technique due to Beurling, for q = oo using a technique due to Lau and Chen, and for 1 < q S oo using amalgam space techniques. Lau's method for q = oo gave the exact value of IIWJJ, while the amalgam space method for q = oo gave an inferior estimate. In this section we prove that W is invertible, and estimate JI w-1 1J for each 1 S q S oo. Again, Beurling's method would suffice for q = l and amalgam ,. 'I ~: spaces for 1 S "": q < oo; instead we generalize a variant of Lau's method to all ,,..,, 1 S q S oo. We prove the surjectivity of W for 1 S q S oo using the same method Beurling used for q = l and Lau for q = oo. i I l .l The following proposition is similar to one proved by Lau and Chen for the ,.... .. special case d = l and q = oo. They did not make use of the minorant w*, but rather assumed that w itself was decreasing on some interval [O, b]. PROPOSITION 4.5.1 [CLl]. Given 1 Sp< oo and 1 S q S oo, and given a nonnegative, even function w on Rd. Then 1/p ( sup II(T) w*(T) ) 11/IIB(p,q) TERd + { (II(A) f ) q/p dA ) l/q S ( lad ~ }Rd lf(t)IP w(At) dt II(A) , + for all measurable functions f; Rd -l- C ( with the standard adjustments if q = 00 ). 185 PROOF: Assume q < oo, the q = oo case being similar. Given b, TE Ri, we compute II(b)w*(b) II(lT) f lf(t)jPdt = ~(~) f lf(t)jPw*(b)dt J[o,TJ J[o,TJ :S II(b/T) f lf(t)IP w*(bt/T) dt, J[o,TJ since bt/T :S b for t E [O, T] and w* is decreasing on Ri. Combining this ,, with similar inequalities for the other quadrants (possible since w is even), j' we have ,, 'l ~ ,...I, : JIiii I' ., 1 ,.' i.' , L ~ ... , :S 2-d IT(b/T) lf(t)IP w*(bt/T) dt a-E0 4 a-[O,Tj = 2-d IT(b/T) f lf(t)IP w*(bt/T) dt )RT :S 2-d IT(b/T) f lf(t)IP w*(bt/T) dt. la" Therefore, (II(b)w*(b))1fp IIJIIB(p,q) - (Lt (I I(b)w*(b) l;TI L, 1/(t)J? dt)"' II~~i)''' < (L. (II(!~T) l_. If( t) 1? '?*( bt/T) dt )"' II~~))"? + 186 / (II(.~) f )q/p d>. )1/q < ( lad ~ }Rd lf(t)IP w(>.t) dt II(>.) ' + where we have made the substitution>. = b/T and used the fact that dT /II(T) is dilation invariant. Taking the supremum over all b E Ri therefore gives the desired inequality. I REMARK 4.5.2. For the case d = l and q = oo, Lau and Chen prove that if suptER+ tw(t) = suptER+ tw*(t) then the constant in Proposition 4.5.1 is best possible. We extend this to higher dimensions as follows. Fix c: > l. It suffices to show that there exists an f E B(p, oo) with llfllB(p,=) = 1 such that sup II(;) r lf(t)IPw(>.t)dt ~ c:C, AE:il.d 2 }Rd + where C = supTERd II(T) w(T). Fix b E (0, 1) C Ri and define + f = (n%))'\,_,,,1? For each j = 1, ... ,d we have 1. Therefore, llfllit.t) dt 2 lad. 0 lr1-6,11 1 f II(>.t)w(>.t) d II(h) lr1-6,1J II(t) t < _1_ f C dt II( h) lr1-6,11 II( 1 - h) C II(l - h) < e:C. EXAMPLE 4.5.3. a. The function tk(t) is continuous on R. If t 2: 1/4 then Therefore tk(t) achieves its maximum somewhere in the interval [O, 1/4]. We I' l ii, compute ,1. ..- i,?,? ,;. 2 sin 21rt . k'(t) ( 21rt cos 21rt - sm 21rt) 7r 2 t 3 and = = sin 21rt . [tk(t)]' k(t)+tk'(t) (41rtcos21rt - sm21rt). 7r 2t2 The maximum of tk(t) therefore occurs at the point b E (0, 1/4) such that tan 21rb = 41rb. There is a unique such point in the interval (0, 1/4); nu- merically, b :=:::: 0.186 and bk(b) ~ 0.461. Since k* = k ? X[o,1; 21, we have suptER+ tk*(t) = bk(b) = suptER+ tk(t). b. Let b be as in part a. Since K(t) = fif k(ti), it follows from part a that 188 COROLLARY 4.5.4. Given 1 ~ q ~ oo, the Wiener transform Wis an injective mapping of B(2, q) into V(2, q), and the inverse mapping w-1 : Range(W) .- B(2, q) is continuous, with -d/2 ( 4.5.1) sup t k(t) ( ) tER+ If q = oo then ( 4.5.1) is equality. PROOF: From Example 4.3.5 and Proposition 4.5.1, I/Wfl/v(2,q) (h. t (n;;) !,)f(t)/' K(>.t)dtt' II~~ir/? 112 > ( sup II(T) K*(T)) l/fl/B(2,q)? TERt From Example 4.5.3b, sup II(T) K*(T) = ( sup t k(t)) d. TER"+ tER+ Therefore Wis injective, and IIW-1 1/ ~ (suptER+ tk(t))-d12, which from Example 4.5.3a is approximately (1.472)d. It follows from Remark 4.5.2 that this is equality if q = oo. I We now complete the proof of the major result of this thesis. THEOREM 4.5.5. Given 1 ~ q ~ oo, the Wiener transform Wis a topological isomorphism of B(2, q) onto V(w, q). PROOF: The combination of Corollary 4.4.3, Proposition 4.4.5, Corollary 4.4.10, and Corollary 4.4.15 establish that the Wiener transform is a con- tinuous linear mapping of B(2, q) into V(2, q) for each 1 ~ q ~ oo. Corollary 189 4.5.4 establishes that W is injective, and that w-1 : Range(W) -+ B(2, q) is continuous for each 1 ~ q ~ oo. It therefore remains only to show that Wis surjective. V Fix any e E V(2, q). Then !:l.J..e E L2 (Rd) for a.e. ,\. Since !:l.J..(t) (-i)d SJ..(t) f:. 0 a.e. (Proposition 4.2.5), we can define a function /J.. by 4.2.2g, "' l .~.. : , :, .~, V V V \: ~ I fl.? . fl.). . /J.. - fl.? . (!:l.J..er . j - ( !:l.?!:l.). er .I I , .,. l . ~ ' ( /:l.J,..!:l.? er V - !:l.J.. . (!:l.?er V V - !:l.J.. ?fl.?? fw V V As fl.? ? !:l.J.. f:. 0 a.e., it follows that /J.. is independent of ,\, and is therefore denoted hereafter by /. Now, V (4.5.2) (!:l.J..e)v (t) !:l.J,..(t) f(t) (-i)d SJ..(t) f(t) IT(-1rit) dJ..(t) f(t). 190 By Proposition 4.2.8, if h E B(2, q) then h ? d'). E L2 (Rd), and (4.5.3) Comparing ( 4.5.2) and ( 4.5.3) we therefore define g(t) = II(-21rit) f(t). Using the Plancherel theorem, the fact that ld').(t)l2 = II(~)2 KPt), and Proposition 4.5.1, we compute (fa., (II~;> l. lg(t)I' K(>.t)dt) .,, II~~i) .,. + Since 11Gllvc2,q) < oo, it follows that g E B(2,q), and therefore Wg E V(2,q). Finally, V A'). W g = 2-d (g ? d').)" = (A'). ? !)" = A').G for a.e. ~, so IIG- W 9llv(2,q) = 0. Since we identify functions in V(2, q) whose difference has zero norm, W g = Gin V(2, q), and therefore Wis surjective. 'I 191 Since the Wiener transform is a topological isomorphism of the Banach space B(2, q) onto the normed linear space V(2, q), it follows that V(2, q) is complete. We prove this in detail in the following corollary. We devote Chapter 5 to proving that V(p, q) is complete for all p, q. COROLLARY 4.5.6. V(2, q) is a Banach space for each 1 ::; q:::; oo. PROOF: Fix 1 ::; q :::; oo. By Theorem 4.5.5, the Wiener transform W is a topological isomorphism of the Banach space B(2, q) onto the normed linear space V(2, q). Assume { Gn}nEZ+ is a Cauchy sequence in V(2, q). a Cauchy sequence in B(2, q). Therefore, w-1 Gn -+ g in B(2, q) for some g E B(2, q). The continuity of W implies then that Gn = ww-1Gn-+ W g I I I in V(2, q), so V(2, q) is complete. I :.?i. ,1111 1 We illustrate now that the value supTERd II(T) w*(T) appearing in Propo- + sition 4.5.1 also arises naturally when amalgam space methods are used. How- ever, the conversion from the continuous norm to a discrete approximation in the proof results, as usual, in an inferior estimate. PROPOSITION 4.5. 7. Given 1 ::; p, q < oo and a nonnegative, even function 192 for all measurable functions f: Rd --+ C. PROOF: Fix any .t) dt )"' n~~) + > L ( (II(!n) E 1 lf(t)jPw*(2m+n+2)dt)q/p ~ n lc2?,2?+1) 2 m o-[2-,2-+i] II(.~) > (log2)d L (:E 11(2m+n-l)w*(2m+n+2) 1 /f(t)IP _i::_)q/p I,l' n m o-(2"' , 2-+11 III(t)l .:: .-1 ,I.I C. ,I = (Jog 2)' ~ ( ~ Il(2m-S) "'*(2m) 11,----? ..--?-?1 If( t ) I' 1n~!)1) .,, .t) dt )"' n~~) ~ 2-d (log2)" LIE Fn,O"(m)P/qr/p n,o- m n,O" 193 Since O < p/q < oo, we have 11 ? lltP/t > 11 ? lltoo. Therefore, ( 4.5.5) L IIFn,o-lltPlt n,o- n,o- = L sup Fn,o-(m) n,o- m > sup L Fn,o-(m) m n,o- ,. '3 i.:.: ..: ,,., (s!p II(2m)w*(2m)) q/p llfllw.(LP,Lt) > (2-d sup II(T)w*(T)) v/p llfllw.(L,.,L.,), Teat the last inequality following from Lemma 3.3.5b. The result follows upon combining ( 4.5,4) and ( 4.5.5). I 194 CHAPTER 5 COMPLETENESS OF THE VARIATION SPACES In this chapter, we prove that the higher-dimensional variation spaces V(p, q) defined in Section 4.3 are complete. Because these spaces are not solid, the completeness is difficult to prove by ordinary techniques. Lau and Chen overcame this difficulty in the one dimensional V(p, oo) case by using helices, a concept developed by Masani. Lau and Chen's proof generalizes immediately to the one dimensional V(p, q) case. We prove the completeness in higher dimensions by using an iterated helix technique. In Section 5.1 we review the basic definitions and properties of helices, first on abstract topological groups and then specifically on the real line. : ,J,, In Section 5.2 we prove that V(p, q) is complete. We review Lau and Chen's ,,, proof for one dimension, then extend it to higher dimensions by using an iterated helix technique. Throughout this chapter, we make use of the symmetric, one-sided, di- rectional, and one-sided directional difference operators defined in Definition 4.2.1. We let e3 = (0, ... , 0, 1, 0, ... , 0) denote the lh unit vector in Rd. We make use of the group representation definitions given in Section 1.9, and use vector-valued integration, following the definitions in [HP). We use the shorthand notation of writing U(t) as Ut for representations U and related maps. 195 Section 5.1. Helices. In this section we define helices and derive their basic properties. We begin with the general definition on abstract topological groups, then turn to the specific case of helices on the real line. The results in this section are taken directly from [Ml] and [M3], and therefore the credit for this section is due to Masani, except for some remarks and examples. We let G denote an arbitrary locally compact abelian group, written addi- tively with identity element O, and let X be an arbitrary Banach space. DEFINITION 5.1.1 [Ml]. A continuous function,: G-+ Xis a variety in X parameterized by G. Given such a variety we define the following terms. <: a. , is a curve in X if G = R. 1 is a surface in X if G = Rd. I I b. Given a, b E G, ,b - 'Ya is a chord of,. .j. c. The chordal length function of, is L,y(a) = 11,a - ,oil for a E G. d. The subspace generated by , is S( 1 ) = span { ,a :a E G}. e. The chordal subspace generated by, is OS(,) = span{,b - ,a a,b E G}. f. 1 is stationary if there exists a strongly continuous unitary representa- tion U of G on M-y such that Ut,a = ,a+t for a, t E G. U is the shift group of,. g. 1 is a helix if there exists a strongly continuous unitary representation U of G on S(,) such that Ut(,b - ,a) = ,b+t - ,a+t for a, t E G. U is the shift group of 1 . Helices will usually be denoted by the symbol h. 196 h. A function c.p: G ---t R is a screw function if it is the chordal length function of some helix in X. 1 A EXAMPLE 5 .1. 2. a. Set d = 1 and let X C L10c (R) be a homogeneous Banach function space. Assume F: R ---t C is such that ~ t F E X for all ). E R, and define Ii: R ---t X by Then Ii is a helix in X, parameterized by R, with shift group {T- .\}.\ER" To see this, first note that since translation is strongly continuous in X. Thus Ii is continuous. Since it remains only to show that {T-.\} is a unitary, strongly continuous repre- sentation of R on X. It clearly is a representation, and the unitarity follows from the fact that X is translation isometric. The strong continuity of the representation follows from the fact that translation is strongly continuous in X, i.e., limb--+a l!Tbg - TaYII = 0 for all g EX. b. Let d = 1 and X = LP(R) and fix F E V(p, q). Then, by part a, Ii.\ = ~ t Fis a helix in LP(R) since LP(R) is homogeneous and~t F E LP(R) for a.e. >.. 197 c. Let d ~ l be arbitrary, and let X C Lf0 c(:R.d) be a homogeneous Banach function space. Fix 1 ~ j ~ d, and assume F: R,d -+ C is such that~{+ FE X for >. E R. Define h: R -+ X by n.i,,,,. _ uA ,,t.+ F - 2l (T- .\e; p - F) . Then, just as in part a, his a helix in X, parameterized by R, with shift group {T-.\e;} .\ER. As in part b, a typical example is formed by taking X = LP(R. d) and FE V(p, q). d. Set d = 2, fix F E V(p, q), and define h.\ ::::: ~t F for >. E R.2 ? Given a, b, >. E R.2 , we compute i.f I '= : ,i,. ?. .i., while 198 Thus, in general, hb+>. - ha+>. f. T_>.(hb - ha), so his not a helix. The same considerations hold for any d > 1, i.e., h>. =~IF is not a helix over Jld when d > 1. LEMMA 5.1.3 [Ml]. . ,,~.. ,,., a. The shift group of a stationary variety I is unique (on S(,)). .~~,,., :u b. The shift group of a helix his unique (on CS(h)). PROOF: We prove only a as bis similar. Assume U, V are two shift groups j ...I for a stationary variety 1 . Ut,a = = Vi,a t G. i,, ,' Then 'Ya+t for all a, E By ?? ' linearity and continuity we therefore have Utf = Vi/ for every f E S( 1 ) = span{,a: a E G}, so U =Von S(,). I LEMMA 5.1.4 [Ml]. Given a helix h, the chordal length function Lt,, is sym- metric, subadditive, and continuous. Further, L1,,(0) = O, and llhb - hall = L1,,(b - a) for a, b E G. PROOF: Recall that L1,,(a) = llha -holl- Lt,, is therefore continuous since his continuous. Given a E G we have Lt,, (a) = 11 ha - ho I/ L1,,(-a), 199 since the shift group U is unitary. Thus h is symmetric. Given a, b E G we compute Therefore, so Lt,, is subadditive and Lt,,(0) = 0. I We turn now to the specific case of helices parameterized by R. We assume . ,.'., ,, . for the remainder of this section that all helices are over R. The following 1111'' .~fl f proposition limits the growth of a screw function. PROPOSITION 5.1.5 [Ml). Given a helix h and a j ER, l ! l ,,. , ,,, PROOF: Set M = max:099 Lt,,(t). Given N E Z+, note that Given a 2: 0 let N = la J, the largest integer N ~ a. Since O ~ a - N < l, If a< 0 then, by the symmetry of Lt,,, 200 PROPOSITION 5.1.6 [Ml]. Given a helix Ii. b. e-t (ho - lit) is Lebesgue-Bochner integrable on R+? PROOF: a. From Proposition 5.1.5, 10C) e-t \\ho - litl\ dt = 10C) e-t Lh(t) dt 00 < Lh(l) 10C) t e-t dt + M 1 e-t dt Lh(l) + M < oo, where M = maxo.. = ~t Fis a ,~-,?.? helix when d = 1, cf., Example 5.1.2a. The existence of the helix average vec- tor allows a Cauchy sequence to be "pulled back" from V(p, oo) to LP, where it will converge. A candidate limit vector for the original Cauchy sequence is then constructed using the fact that helix chords are determined by the aver- age vector (Proposition 5.1.11). Masani's results on helices parameterized by R, Proposition 5.1.5 through Proposition 5.1.11, are the critical facts which make Lau and Chen's proof possibile. As mentioned in Remark 5.1.12, Masani's results on helices over R can be extended to helices over Rd. However, by Example 5.1.2d, h>.. = ~t Fis not a helix when d > l. Therefore, helices over Rd are not appropriate for proving the completeness of V(p, q) in higher dimensions. Instead, we use the fact that n>..; = ~{,: Fis a helix over R for each j = 1, ... , d, and that ~ t = ~ l~ ? ? ? ~ i1. An iterated averaging technique allows us to "pull back" a Cauchy sequence from V(p, q) to LP. An iterated chord reconstruction then gives the candidate limit vector for the Cauchy sequence. 206 . ; THEOREM 5.2.1 [CLI]. Given a homogeneous Banach function space X C Lf c('R) and given l :=:; q :s; oo. Assume cp:R+---+ R+ satisfies >i.e-A/cp(>i.) E 0 Lq' (R+), where this space is taken with the Haar measure d>i./>i. for R+? Let Y be the space of functions F such that Then Y is a Banach space, once we identify functions F, G E Y such that 1/F-Gl/y = 0. PROOF: The seminorm properties of II ? IIY are evident, so Y is a normed linear space once we make the identification of functions whose difference has zero norm. It remains to show that Y is complete. Assume that { Gn}nEZ+ is a Cauchy sequence in Y. a. Given n E Z+, define Ii": R---+ X by By Example 5.1.2a, Ii" is a helix in X, parameterized by R, with shift group {T-AhE11.? This helix has an average vector defined by 100 00 an = e-A (n; - Ii~) d)i. = -1 e-A Lit Gn d>i.. By Proposition 5.1.6, an E CS(h") C X. The sequence {an} is Cauchy in X since 11am - anllx 00 111 e-A Lit(Gm - Gn)d>i.llx 207 ) } Jo/o o ). .\ + d). < e- Ill\.\ (Gm - Gn)l/x T ~ (f \ Q(A)? ll~t(Gm - G.)llk ~A),,. ([ I~ c:; I" ~A) 1/r' = CI/Gm - GnllY --+ 0 as m, n --+ oo (the cases q = I, oo are similar). Therefore, an--+ a {or some a EX. By Proposition 5.1.11, (5.2.1) = T_.\an - Toan - L.\ T_.an ds = 2L\! an - L.\ T_.ands. Since a EX C Lfoc(:R), we can define We compute (5.2.2) L\f G(-y) = ?[ G(-y + >.) - G(-y)] = a(,+>.) - L-,+J.. a(s) ds - a(,) + L"' a(s) ds -,+J.. - a(,+>.) - a(-y) - 1.., a(s)ds 208 J , = 2~ta(,) - L)t. a(, +s)ds = (2~ta - l)t. T_.ads)(,). From (5.2.1) and (5.2.2), ll~t(G - Gn)llx = //2~f(a - an) - L)t. T_.(a - an)dsl/x $ /IT-}t.(a - an)Jlx + Ila - anllx + 1). jjT_.(a - an)Jlx ds = (2 + A) Ila - anllx- Therefore, for each :fixed A, (5.2.3) --+ 0 as n --+ oo. b. We show now that GEY. Define By definition, G E Y if and only if f3 E Lq(R+)- Now, Gn E Y, so f3n E Lq(R+ ). By the triangle inequality and (5.2.3), (5.2.4) Moreover, 209 (5.2.5) L00IIPm - Pnll: = cp(A)'1 111~tGmllx - u~tGmllxlq ~).. ~ L00 cp(>-.)q u~t(Gm - Gn)ll1 ~).. = I/Gm - Gnl/i --+ 0 as m, n --+ oo. Thus {Pn} forms a Cauchy sequence in Lq(R+), so must converge to some element of Lq(.R+)- Since Pn--+ p pointwise by (5.2.4) we must have Pn--+ P in Lq(.R+)- Thus p E Lq(R+), so GEY. c. We show now that Gn --+ G in Y for the case 1 < q < oo. Since Bn(>-.) = cp(>-.) 1/~t(G - Gn)llx, it suffices to show that 1/Bnl/q--+ 0 as n--+ oo. By (5.2.3), On --+ 0 pointwise, and, by (5.2.4) and (5.2.5), Pn --+ P both pointwise and in Lq(.R+ ). Also, 8n ~ Pn + P, so Thus 2q P(>-.)q + 2q Pn(>-.)q - Bn(>-.)q ~ O, so we can apply Fubini's thoerem in the following calculation: = 9 oo p(>-.)9 d).. + 2q Loo P(>-.)9 d).. Loo d).. 2 L - - - limsup Bn(>-.)9 -. O ).. 0 ).. n-+oo O ).. 210 fore Gn -+ G in Y. d. Finally, we show that Gn -t G in Y for the case q = oo. Fix e > 0. Then there exists an N > 0 such tha.t Also, by definition of II ? IIY when q = oo, there must exist a.~ > 0 such that From (5.2.3), there then exists an M 2'.: N such that 'f'(~) ll.6.t(G - Gn)llx ~ e for n ~ M. Therefore, < 4e for n ~ M. Thus Gn -t Gin Y. I EXAMPLE 5.2 .2. a. Set X = LP(R), ./. E Lq'(R+), so Y is complete. Evaluating, IIFlly - ll'f'(A) ? lldt FIIL?(i.) ''L?(B.+) - (t G1 : ILl,;2Fh)I' d-r)' /p ~~ )". - (f( X 1 : ILl,F(-r)I' d-yt ? ~ )". - IIFllv(p,q), so V(p, q) is complete (ford= 1). We now extend Theorem 5.2.1 to higher dimensions. THEOREM 5.2.3. GivenahomogeneousBanachfunctionspaceX C .Lf c(Rc1) 0 and given 1 ~ q ~ oo. Assume .)/c,o(A) E 212 Lq' (Ri), where this space is taken with the Haar measure d>./II(>.) for ft'.!-. Let Y be the space of functions F such that Then Y is a Banach space, once we identify functions F, G E Y such that !IF - Gj[y = 0. PROOF: The seminorm properties of II ? jjy are evident, so Y is a normed linear space once we make the identification of functions whose difference has zero norm. It remains to show that Y is complete. Assume that { Gn}nEZ+ is a Cauchy sequence in Y. Fix n E Z+, let aon = 1 A Gn, and define Ii n: R-+ X by for >.1 E R. By Example 5.1.2c, 1i1n is a helix in X, parameterized by R, with shift group U 1 = {T-~1 e1 h ER.? This helix has an average vector a1n 1 defined by By Proposition 5.1.6, a1n E CS(1i1n) C X. Since X is closed under translations, ~i~ a 1n EX for >. 2 E R. Therefore is also a helix in X, parameterized by R, with shift group U2 = {T-~2 e2 h ER.? 2 This helix has an average vector a2n defined by 00 2 2 a2n = - Jo/ e- ~, A + d' E CS(.z, n) C X u~2 a1n "'2 " . 213 J I Continuing in this way we obtain helices li1n, ... , lidn and average vectors a1 n, , , . , Qdn such that and Define -.. = -100 e-.\? d~! a(d-I)n dAd = -100 e-.\? d~; (- Loo e-.\?- 1 d~~=:)+ a(d-2)n dAd-1) dA,1. 100 (-1)2 Loo e-.\? e-.\?- 1 d1; d~~=!)+ O:(d-2)n dAd-1 dAd 00 - (-l)d [ ... Loo e-.\? ???e-.\1 d~; .. ,di1aondA1 ... dAd = (-l)d f II(e-.\)dIGndA. lat Then 11am - anllx = 11.l_., II(e-.\)dI(Gm - Gn)dAllx + < JIj_ ? Il(A e -.\ ) !Id.+\ (Gm - Gn )II x Ild(AA ) + $ /l..; ac;-1)n - >..; - -1:.in -1:.in - ''>..; no 1 - T , "",?n - a ?n - { >.. T-a,? e; a 3?n ds3? - _,.1e1 ..... 3 lo0 Therefore ' (5.2.6) AfGn = ~A>.d..+, A 1+ ???.?.l,>..1aon - ~A>d...+, ? ? ? A 2+ (2 A 1+ .U.>..2 .U>,.1 - (2 Al"; - t? T_.,., ds,) A~?;?? -A~"; "'1? = ( 2A l;" - t? T_.,., d?1) ?. . (2 A'J:; - t? T-???? ds,) "'?? Since a E X C L1 (Rd) we can, by Fubini's the0rem, define loc ' Fo(1) == a(,y) F,(-y) = 2 (Fn(-Y) - f."' Fo(si,-Y2,???,"Y?)cls1), 215 , > We compute ~{: F;(-r) 1 .,., , rrJ+}l,; F;-1(-r+>i;e;) - Jo F,_1 (-y1, ... ,-y,_1,s,,-y;+1,???,'Yi)ds; /'Yi - F;-1(-r) + Jo F;-1("Y1,???,"Y;-1,s,,"Y;+1,?? ? ,"Yi)ds; "f;+}l,; - L F,-1 (" "(1, ??? , 'Yi-I, s;, "Y;+I, ... , 'Yi) ds; ,' , ??'Yi '' j, ' Therefore, just as in (5.2.6), (5.2.7) .1.f G = (2 n.{; - [' T_.,,, ds, )-- ? (2 .1.~"'; - t? T_.,., ds,) a. Define Ho = a - an and, for j = 1, ... , d, 216 J I Then, (5.2.8) IIH;llx = //(26{; -1>.1 T-.1e1 ds;)H;-1//x = (2 + >.;) IIH;-1 llx- Combining (5.2.6), (5.2.7), and (5.2.8), .,,,, I J ll ?' ,, ,, ,. , ..., .. ,.f.,~ . ,, "' = II(2 +>.)Ila - an/IX? Hence, for each fixed >., ~ II(2 + >.) cp(>.) jja - anl/x --+ 0 as n--+ oo. The remainder of the proof is now precisely similar to parts b, c, and d of the proof of Theorem 5.2.1. I 217 I ?.?.? PART IV .... WAVELET THEORY J, i 218 CHAPTER 6 FRAMES Frames were invented by Duffin and Schaeffer in their work on nonharmonic Fourier series as an alternative to orthonormal bases in Hilbert spaces, cf., [DS]. They were later used by Daubechies, Grossmann, and Meyer to formu- late wavelet theory in ? 2 (R), cf., [DGM; D1]. Grochenig has extended the notion of frames ( and the related concept of sets of atoms) to Banach spaces, cf., [G]. This chapter is an essentially expository review of basic results on frames and sets of atoms, especially in Hilbert spaces. We have combined results from many sources, including [D1; DGM; DS; G; GK; Y], with re- marks, examples, and minor results of our own, into a single survey chapter. In Section 6.1 we recall the definitions and basic properties of bases in Banach and Hilbert spaces. In Section 6.2 we define frames for Hilbert spaces and discuss their basic properties. The primary result is that given a frame {xn}, any element x E H can be written as :z: = ~ CnXn, where the scalars {en} are explicitly known ( although not necessarily unique), and the series converges unconditionally. In Section 6.3 we characterize those frames which are bases, i.e., those frames for which the representations :z: = ~ Cn:Z:n are unique for all :z:. In Section 6.4 we discuss sets of atoms, which are a dual concept to frames. The term atoms is an unfortunate terminology, since this word is heavily 219 overused in the literature. In particular, the sets of atoms discussed here are not related to the atoms and atomic decompositions appearing in Littlewood- Paley theory. We discuss in this section the exact relationship between frames and sets of atoms, and show that, while atoms are a more general concept, in most practical applications atoms and frames in Hilbert spaces are equivalent. In Section 6.5 we discuss the formulation of frames and sets of atoms in Banach spaces. Finally, in Section 6.6 we prove a stability result for sets of atoms in Banach spaces. In particular, we prove that the elements of a set of atoms may be perturbed by a small amount without destroying the atomic properties. 220 Section 6.1. Bases. In this section we review the basic definitions and properties of bases in Banach and Hilbert spaces. DEFINITION 6.1.1. Given a sequence {xn}neZ+ of elements of a Banach space X. a. The span of {xn}, denoted span{xn}, is the set of finite linear com- binations of elements of {xn}- The closed linear span of {xn}, denoted .. ? span{xn}, is the closure in X of span{xn}- b. {xn} is complete if span{xn} = X, or, equivalently, if? E X' and ?(xn) = 0 for all n implies?= 0. c. {xn} is minimal if Xm-=/:- span{xn}n~m for each m. d. { x n} is a basis if for each x E X there exist unique scalars an ( x) such that x = I;an(x)xn. The basis is unconditional if the series I;an(x)xn converges unconditionally for each x, cf., Section 1.4. The basis is bounded if O < inf llxnll ~ sup llxnll < oo. REMARK 6 .1. 2. a. Bases are complete and minimal, but the reverse need not be true. b. Every basis is a Schauder basis, i.e., each coefficient functional an is continuous and therefore an element of X'. c. If {xn} is a basis then {xn} and {an} are biorthonormal,i.e., am(xn) = Dmn. The following proposition states that the existence of a biorthonormal sequence is equivalent to minimality. 221 PROPOSITION 6.1.3 [SJ. Given a sequence {xn}nEZ+ in a Banach space X. a. { xn} is minimal if and only if there exists a sequence { an} C X' which is biorthonormal to {xn}- b. { xn} is minimal and complete if and only ifthere exists a unique sequence {an} C X' which is biorthonormal to {xn}- PROPOSITION 6.1.4 [SJ. Given a complete sequence {xn}nEZ+ in a Banach space X with every Xn -=f- O, the following statements are equivalent. a. { xn} is an unconditional basis for X. b. There exists C 1 > 0 such that for all scalars c1, .. . , CN and all signs 0-1, ??? ,0"N = ?1, c. There exists C 2 > 0 such that for all scalars b1, ... , bN and c1, ... , c N d. There exist C 3 , C4 > 0 such that for all scalars ci, ... , CN, N C4 IIL lcnl Xnll? 1 DEFINITION 6.1.5. Two bases {xn} and {Yn} for a Banach space X are equivalent if there exists a topological isomorphism U: X -+ X such that 222 l I U Xn = Yn for all n, or, equivalently, if E CnXn converges if and only if E CnYn converges. We list some additional facts about bases in Hilbert spaces. The inner ' product in a Hilbert space His written (?, ?}. DEFINITION 6.1.6. A sequence {en} of elements of a Hilbert space His an orthonormal basis if a. { en} is orthonormal, i.e., (em, en} = 5mn, .,. b. the Plancherel formula holds, i.e., E l(x,en}l 2 = llxll 2 for x EH. All orthonormal bases are bases (in the sense of Definition 6.1.1), with x = I: {x , en} en for x E H. DEFINITION 6.1. 7. A basis for a Hilbert space H is a Riesz basis if it is equivalent to some orthonormal basis for H. PROPOSITION 6.1.8 [Y; GK]. Given a sequence {zn}nEZ+ in a. Hilbert space H, the following statements are equivalent. a. { Xn} is a. Riesz basis for H. b. {xn} is a. bounded unconditional basis for H. c. {zn} is a. basis for H, and d. {zn} is complete and there exist A, B > 0 such that for all sea.la.rs 223 ti, N N N AL lc..12 < ll:Ec..zn/1 < B Llc..12 - 1 1 1 The following is known as Orlicz' Theorem. PROPOSITION 6.1.9 [O; LT; SJ. Given a sequence {zn} in a Hilbert space H. If .E Zn converges unconditionally then ~ llznll2 < oo. The converse of Proposition 6.1.9 is not true. 1? 224 6.2 Frames in Hilbert Spaces. In this section we define and describe the basic properties of frames in Hilbert spaces. DEFINITION 6.2.1. A sequence {zn}neJ in a Hilbert space His a frame if there exist A, B > 0 such that for all z E H, (6.2.1) Ajjzll2 < L l{z,zn)l2 < B llzl12. nEJ The numbers A, B are the frame bounds, A being the lower bound and B the upper bound. The frame is tight if A= B. The frame is exact if it ceases to be a frame whenever any single element is deleted from the sequence. REMARK 6.2.2. a. Asequence{xn}forwhich}:j(z,zn)l2 < ooforallz EH is a Bessel sequence (cf., [Y]). By the Uniform Boundedness Principle, a Bessel sequence will possess an upper frame bound B > O, i.e., L l(z, zn)l2 ~ Bllzll 2 for z E H. In applications, a sequence which is a frame is often easily shown to be a Bessel sequence, while the lower frame bound is more difficult to establish. b. From the Plancherel formula, every orthonormal basis is a frame with A = B = 1. Any orthonormal sequence which satisfies the Plancherel formula is an orthonormal basis, and therefore gives a decomposition of the Hilbert space in terms of the basis elements. The pseudo-Plancherel formula (6.2.1) for frames also implies a decomposition in terms of the frame elements, al- though the representations induced need not be unique (Proposition 6.2.Sc). 225 c. Since LI (x, xn) l2 is a series of nonnegative real numbers, it converges absolutely, hence unconditionally. That is, every rearrangement of the sum also converges, and converges to the same value. Thus, every rearrangement of a frame is also a frame, and all sums involving frames converge uncondi- tionally. Therefore, we can use any countable index set to specify a frame. For this reason we suppress the index set in the remainder of this chapter. d. Frames are complete, for if x E H and (x, xn) = 0 for all n, then A /lxll 2 2 ::; L l(x,xn)l = O, so x = 0. Therefore, any Hilbert space which possesses a frame must be separable, for the set of finite linear combinations of { xn}' with rational coefficients (i.e., rational real and imaginary parts) is a countable dense subset of H. Every separable Hilbert spaces does possess frames since it possesses orthonormal bases. e. Frames were introduced in 1952 by Duffin and Schaeffer in connection with nonharmonic Fourier series [DSJ. Much of the general theory of frames was laid out in that paper, although frames were apparently not used in any other context until the paper [DGMJ by Daubechies, Grossmann, and Meyer. The following example shows that tightness and exactness are not related. EXAMPLE 6.2.3. Given an orthonormal basis {en}nEZ+ for a Hilbert space H. a. { en} is a tight exact frame for H with bounds A = B = 1. b. { e1, e1, e2, e2, e3, e3, ... } is a tight inexact frame with bounds A = B = 2 but is not orthogonal and is not a basis, although it contains an orthonormal basis. 226 ~-- ' I , c. { e1, e2/2, e3/3, ... } is a complete orthogonal sequence and is a basis, but is not a frame. with bounds A= B = 1, and no nonredundant subsequence is a frame. e. {2e1, e2, e3, ... } is a non tight exact frame with bounds A = 1, B = 2. EXAMPLE 6.2.4. The frames used in wavelet theory (e.g., [DGM]) are co- herent state frames, i.e., they are generated from a single fixed element by the action of a group representation. Precisely, they have the form {U-y,.g}, where g E H is fixed, U is a representation of a locally compact group G on H, and {,n} C G. Typically, 'Yn will be a regular lattice of points in G, though this is not necessary. For example, in Chapter 7 we discuss the situation H = L 2(R d), G is the Heisenberg group, U is the Schroedinger representation, and U-y,,.,.9 = TnaEm1,g form, n E zd. The structure inherent in coherent state frames provides a means for an- alyzing them. For example, assume that G is compact, U is unitary and square-integrable, and {U-y,.9}neJ is a Bessel sequence in H. By definition of square-integrability, there then exists an admissible vector/, i.e., an element /EH such that fa /(U-yf,f)/ 2 d, < oo, where d, is the left Haar measure on G. Since {U-y,. g} is a Bessel sequence, we therefore have LL /(U-yf, U-y,.9)1 2 d, = LL /(U-yf, U-y,.g)j 2 d, ~ LB 2 //U-yf// d, 227 ..... = B 11111 2 IGI, Note that IGI < oo since G is compact. By Proposition 1.9.1, independent of n. Therefore J must be finite, and hence H must be finite- dimensional. We now prove some basic properties of frames. Part a of the following lemma is proved in [DS]. LEMMA 6.2.5. Given a Bessel sequence {xn} with upper bound B. a. ~ CnXn converges unconditionally in H for every { en} E l 2 , and b. Define Ux = {(x,xn)} for x EH. Then U:H 2 -t f. continuously, and its adjoint U*:l2 -t His given by U*{cn} = }: cnXn, c. If { xn} is a frame then U is injective and u? surjective. PROOF: a. Let F be any finite subset of the index set J. Then (6.2.2) 2 sup l(L CnXn, Y)l 111111=1 nEF sup IL Cn (xn,Y}j2 111111=1 nEF 228 < (I: 2sup icni ) (I: l(xn,Y}1 2 ) IIYll=l nEF nEF < sup (I: lcnl 2 ) BIIYll 2 IIYll=l nEF BI: lcnl 2 , nEF Since I: icnl 2 converges absolutely and unconditionally, it follows from (6.2.2) that I: CnXn converges unconditionally in H, cf., Lemma 1.4.2c. Therefore we can replace F by J in (6.2.2), i.e., II I: Cn:Z:n 11 2 s; BI: icnl 2 ? b. That U is well-defined and continuous follows from the definition of Bessel sequence, for IIUxll~ = I: l(x,xn}l 2 s; B 11:z:ll 2 , Its adjoint U*:f2 ---+ H is therefore well-defined and continuous, so we need only verify that it has the correct form. If { en} E f 2 then I: CnXn converges to an element of H by part a, so given :z: EH we can compute (x, U*{cn}} - (Ux,{cn}} c. Follows from the fact that frames are complete. I PROPOSITION 6.2.6 [DS]. Given a sequence {xn} in a Hilbert space H, the following statements are equivalent. a. {xn} is a frame with bounds A, B. 229 b. Sx = I::(x,xn)Xn is a bounded linear operator with AI :s; S :s; BI. In case these hold, the series in b converge unconditionally. PROOF: b ? a.If b holds then (Aix,x) :s; (Sx,x) :s; (Bix,x) for x EH. As (Jx,x) = jjxjj 2 and (Sx,x) = I: j(x,xn)l2 , it follows that {xn} is a frame. a ? b. Assume {xn} is a frame and fix x EH. Then I::l(x,xn)l 2 < oo, so Sx = I::(x,xn)Xn converges unconditionally by Lemma 6.2.5. The lemma also implies that jjSxjj 2 :s; BI: l(x,xn)l 2 :s; B 2 jjxll 2 , so Sis bounded with jjSjj :s; B. The relations AI :s; S :s; BI follow immediately from the definition of frame. I I !, DEFINITION 6.2. 7. Given a frame { xn}, the operator Sx = I::(x, xn)Xn is the frame operator for {xn}? From AI :s; S :s; BI it follows that A jjxjj :s; jjSxjj :s; B jjxjj for x E H. S is therefore continuous and injective, and s-1 : Range( S) ----+ H is continu- ous. The following proposition shows that Sis surjective, hence a topological isomorphism of H. PROPOSITION 6.2.8 [DS]. Given a frame {xn}. a. S is invertible and B-1 I :s; s-1 :s; A-1 I. b. { s-1 xn} is a frame with bounds B-1 , A- 1 . c. Given x EH, and these series converge unconditionally. 230 PROOF: a. Note that O 5 I - B-1 S :::; I - jI = ?I since AI < S < BI. Therefore III - B-1s11 < ll 8 iAIII = B;A < 1, whence B-1s, and therefore S, is invertible. The operator s-1 is positive since As s-1 commutes with both I and S we can therefore multiply through by s-1 in the equation AI 5 S 5 BI, obtaining B-1 I 5 s-1 5 A-1 I, cf., [Heu, p. 269]. b. The operator s-1 is self-adjoint since it is positive. Therefore, = s-l (I: (S-l z, Zn} Zn) ..,...,. "' That {S-1zn} is a frame now follows from part a and Proposition 6.2.6. c. We compute and The unconditionality of the convergence follows from the fact that { Zn} and {S-1zn} are both frames. I 231 DEFINITION 6.2.9. Given a frame {zn} with frame operator S, the frame {S-1 zn} is the dual frame of {zn}. REMARK 6.2.10. a. The expressions in Proposition 6.2.Bc are what we mean when we informally say that a frame {zn} gives a decomposition of the Hilbert space. b. In case {zn} is a tight frame, i.e., A= B, the conclusions of Proposition 6.2.8 reduce to s = AI, s-1 = A-1 I, and z = A-1 :E (z, Zn)Zn for z E H. We now prove some results relating to the uniqueness of the decomposition given by a frame. The following proposition shows that the scalars given in Proposition 6.2.8c have the minimal l.2 norm among all choices of scalars { en} for which z = I; CnZn? PROPOSITION 6.2.11 [DS]. Given a frame {zn} and given z E H. If z = E CnZn for some scalars {en}, then PROOF: Define an= (z,s-1 zn); then z = :EanZn by Proposition 6.2.Bc. Since E Ian j2 < oo, assume without loss of generality that I; lcnl2 = oo. Then (z, s-1 :c) - \L an:cn, s-1z) - Lan (S-1 zn,:c) 232 - Lanan ({an}, {an}) and (x, s-1 x} - \L CnXn, 5-lx) - L Cn (s- 1 xn, x} - LCnan = ({en}, {an}). Therefore { Cn - an} is orthogonal to {an} in l 2, whence PROPOSITION 6.2.12 [DS]. The removal of a vector from a frame leaves either a frame or an incomplete set. Precisely, (xm, s-1 xm} = 1 ? {xn}n#m is incomplete. PROOF: a. Fix m and define an = (xm, s-1xn}- By Proposition 6.2.Sc, Xm = E anXn, However, Xm = E c5mnXn as well, so by Proposition 6.2.11, n n n n#m n#m 233 Therefore, b. Suppose that am = l. Then Ln#m lanl 2 = o, so an = (s- 1 xm, Xn) = 0 for n =I=- m. Thus s-1 xm is orthogonal to Xn for n =I=- m. However, s-1xm =I=- 0 since (S-1 xm,xm) =am= 1 =/=- 0. Therefore {xn}n#m is incomplete in this case. c. On the other hand, suppose am =/=- l. Then Xm = l-~m Ln#m anXn, so for x EH, where C = 11 - am 1-2 Ln#m lanl 2 ? Therefore, n whence Thus {xn}n#m is a frame with bounds A/(1 + C), B. I In the course of the proof of Proposition 6.2.12 we proved the following. COROLLARY 6.2.13. Given a frame {xn} and given m, L 1 2 1 - l(xm, s-1 xm)l 2 - 11 - (xm, s-1 xm)l2 l(xm,s- xn)l 2 n#m 234 In particular, if (xrn, s-1 x,n) = 1 then (xrn, s-1 xn) = 0 for n -=I=- m. COROLLARY 6.2.14. Given a frame {xn}, the following three statements are equivalent. a. { Xn} is exact. b. {xn} and {S- 1 xn} are biorthonormal. c. (xn, s-1 xn) = 1 for all n. PROOF: a::::} c. If { Xn} is exact, then, by definition, { Xn}n:;t:rn is not a frame for any m. Therefore, by Proposition 6.2.12, (xrn, s-1 xrn) = 1 for every m. c ::::} a. If (xrn, s-1 xrn) = 1 then {xn}n:;t:rn is not a frame by Proposition 6.2.12. By definition, {xn} is exact if this is true for all m. c ::::} b. Follows from Corollary 6.2.13. I COROLLARY 6.2.15. Given a tight frame {xn} with bounds A B, the following statements are equivalent. a. { Xn} is exact. b. { xn} is an orthogonal sequence. c. llxnll 2 = A for all n. PROOF: Follows from Corollary 6.2.14 and the fact that S = AI. I PROPOSITION 6.2.16. a. Frames are norm bounded above, with sup llxn!l 2 ~ B. b. Exact frames are norm bounded below, with A~ inf llxnll 2 ? 235 PROOF: a. Fix m; then n b. If {zn} is an exact frame then {zn} and {S-1 zn} are biorthonormal by Corollary 6.2.14. Therefore, for m :fixed, A lls- 1 .".,' mll 2 < ~L.J I(s-1 )12 Zm,Zn n As { :en} is exact we have Zm -=/= O, so s-1zm -=/= 0 and the result follows. I REMARK 6.2.17. Example 6.2.3d shows that inexact frames need not be bounded below. We collect now some remarks on the convergence of E CnZn for arbitrary sequences of scalars. EXAMPLE 6.2.18. In general, it is not true that :z: = E CnZn implies that E lcnl2 < oo. For example, let {zn} be any frame which includes infinitely many zero elements and talce the coefficients of the zero elements to be 1. Less trivially, let {en}neZ+ be an orthonormal basis for Hand define In = n-1 en and 9n = (1 - n-2 ) 112 en, Then {ln,9n} is a tight frame with A = B = 1. Now consider the element z = E n-1 en; we have :z: = E (l ?In+ 0 ? 9n) while 236 PROPOSITION 6.2.19. Given a frame {:t:n} which is norm bounded below, ~ lcnl 2 < oo {:} ~ Cn:t:n converges unconditionally. PROOF: Assume I: Cn:t:n converges unconditionally. Then, by Proposition 6.1.9, I: lcnl2 llznll 2 = I: llcn:t:nll 2 < oo. Since {:t:n} is norm bounded below it follows that I: !cnl2 < oo. The converse is Lemma 6.2.5. I EXAMPLE 6.2.20. There exist frames {:t:n} which are norm bounded below and scalars { en} such that I: Cn:t:n converges but I: lcnl2 = oo. Let { en}nEZ+ be an orthonormal basis for H, and consider the frame {e1,e1,e2,e2,???}, which is norm bounded below. The series (6.2.3) converges strongly to 0. However, the series does not converge. Therefore the series (6.2.3) converges conditionally, cf., Lemma 1.4.2e. Since {n-112} ~ P.2, the conditionality of the convergence also follows from Proposition 6.2.19. 237 Section 6.3. Frames and bases. In this section we determine the exact relationship between frames and bases in Hilbert spaces. PROPOSITION 6.3.1. Inexact frames are not bases. PROOF: Assume {xn} is an inexact frame, with frame operator S. Then, by definition, {xn}n:;cm is a frame for some m, and is therefore complete, while no subset of a basis can be complete. In particular, define an= (xm,S-1 :z:n)i then ~ c5mnXn = Xm = I; anXn (Proposition 6.2.8c). Since am -=/- 1 by Proposition 6.2.12, these are two different representations of Xm? I LEMMA 6.3.2. Frames a.re preserved by topological isomorphisms. Precisely, we have the following. Let H1 , H2 be Hilbert spaces, and let {xn} be a frame for H1 with bounds A, B and frame operator S. Assume T: H1 -+ H2 is a topological isomorphism. Then {Txn} is a frame for H2 with bounds AIIT-1 1!-2 , B IITll 2 and frame operator TST*. Moreover, {Txn} is exact if and only if {xn} is exact. PROOF: First note that for each y E H2, By Proposition 6.2.6, it therefore suffices to show that A I/T-1 11-2 I:$ T ST* :$ B IITll 2 I. Given y E H we have (T ST*y, y) = (S(T*y), (T*y)), so (6.3.1) A I/T*yl/ 2 < (T ST*y, y) < B IIT*yl/ 2 , 238 since AI :s; S :s; BI. Since T is a topological isomorphism, (6.3.2) IIYII JIT-111 Combining (6.3.1) and (6.3.2), Allyll2 ? IIT-1112 :s; (T ST Y, y) :s; B IITll2 IIYll2, as desired. The statement about exactness follows immediately from the fact that topological isomorphisms preserve complete and incomplete sequences. I The statement and a different proof of the following can be found in [Y]. PROPOSITION 6.3.3. A sequence {xn} in a Hilbert space His an exact frame l if and only if it is a bounded unconditional basis. !: ' PROOF: =}. Assume {xn} is an exact frame. Then {xn} is bounded in norm by Proposition 6.2.16. By Proposition 6.2.Sc, x = I:(:z:, s-1 :z:n) :Z:n for all :z:, and this series converges unconditionally. This representation is unique, for if :z: = E Cn:Z:n then since {:z:n} and {S-1:z:n} are biorthonormal (Corollary 2.3.14). Thus {:z:n} is a bounded unconditional basis. {::::. Assume {:z:n} is a bounded unconditional basis for H. Then by Propo- sition 6.1.8, { :z:n} is equivalent to an orthonormal basis for H, i.e., there exists an orthonormal basis { en} and a topological isomorphism U: H -4 H such that U en = :Z:n for all n. Since { en} is an exact frame, { :z:n} must also be an exact frame by Lemma 6.3.2. I 239 REMARK 6.3.4. We can exhibit directly the topological isomorphism U used in the proof of Proposition 6.3.3. First note that 5-1/ 2 exists and is a positive topological isomorphism of H since both 8 and 5-1 are positive topologi- cal I? somorphisms, e.g., [We, Theorem 7.20}. Since {zn} is exact, {zn} and {s-1 } :Z:n are biorthonormal. Therefore, (s-112 Zm, 8 -1/2 Zn ) = (Z m, 8 -1128 -112 Zn) = ( Zm, 8 -1 Zn ) = Ocm n? Thus { s-112:cn} is orthonormal. It is complete since topological isomorphisms Preserve complete sequences. Thus, {8-112:cn} is an orthonormal basis for ll, and the topological isomorphism U = 8 112 maps this orthonormal basis onto the frame {zn}, For inexact frames, {5-1/2zn} will not be an orthonormal basis, but will ,? f he a tight frame. COROLLARY 6.3.5. Any frame in a Hilbert space is equivalent to a tight 1 f.razne. Precisely, if {zn} is 8 frame with frame opera.tor 8 then 5- 2 / is a Positive topological isomorphism of H and {5-1/2zn} is a tight frame with bounds A = B = 1. PROOF?? It tio 1 1 1 2 ows fr om Le mma 6 . 3 ?2 that {s- 1 zn} is a frame. Since the frame is tight by Proposition 6.2.6. I 240 EXAMPLE 6.3.6. From Propositions 6.1.8 and 6.3.3, if {xn} is an exact frame then (6.3 .3) L CnXn converges # L CnXn converges unconditionally. Now let { en}nEZ+ be an orthonormal basis for H, and consider the frame {xn} = {e1,e1,e2,e3, ... }. The series LCnXn will converge if and only if L lcnl2 < oo since {xn} is obtained from an orthonormal basis by the ad- dition of a single element. Since {xn} is norm bounded below, it follows from Proposition 6.2.19 that lcnl2 L < oo if and only if L CnXn converges unconditionally. Therefore (6.3.3) holds for this nontight, inexact frame. 241 Section 6.4. Atoms in Hilbert spaces. By definition, a sequence { Xn} is a frame if there exists a norm equivalence between llxllH and II{ (x, xn)}//l 2 (Definition 2.3.1). Given such a frame, it follows that there exist coefficients { an( x)} such that x = Lan( x) Xn, in particular, an(x) = (x,S-1:z:n) (Proposition 6.2.8). Since {S-1:z:n} is also a frame, there is also a norm equivalence between 1/x/lH and //{an(x)}//l 2 ? In this section, we examine a dual concept to frames, due to Grochenig, which begins from the existence of coefficients { an( x)} which reproduce x and satisfy a norm equivalence. We establish in this section the exact relationship between frames and Grochenig's sets of atoms, in the Hilbert space setting. DEFINITION 6.4.1 [G]. Given a sequence {xn} in a Hilbert space H, and given a sequence { an} of linear functionals on H. If a. x = L an(x) Xn for every x EH, b. there exist constants A, B > 0 such that for each x E H, then {xn; an} is a set of atoms for H. A, Bare the atomic bounds, and the functionals {an} are the atomic coefficient functionals. REMARK 6.4.2. a. We do not assume that the representation x = L an(x) Xn in Definition 6.4.1 is unique, i.e., {xn} need not be a basis for H. b. Since !am(:z:)1 2 ~ 1: jan(x)j2 ~ B llx112, each functional am is continuous, and is therefore given by the inner product with a unique Ym E H, 1.e., 242 am ( ?) = (? , Ym). We identify the functional am with the element Ym, and refer to {yn} as the atomic coefficients. c. If {:z:n} is a frame, then {:en; s-1:z:n} is a set of atoms by Proposition 6.2.8, where Sis the frame operator for {:en}- We establish a partial converse to this result in this section (Proposition 6.4.5). From Definition 6.4.1 and Remark 6.4.2b we immediately obtain the fol- lowing. PROPOSITION 6.4.3. If {:i:n; Yn} is a set of atoms with atomic bounds A, B then {Yn} is a frame with frame bounds A, B. EXAMPLE 6.4.4. It need not be true that {:en} is a frame for H if {:z:n; Yn} is a set of atoms. For example, if { en}nEZ+ is an orthonormal basis for H then ,. ,,., { en, nen} is not a frame since it is not bounded in norm. However, it does .. form a set of atoms for H if we define the atomic coefficents to be { en, O} or PROPOSITION 6.4.5. Given a set of atoms {:z:n; Yn}, with atomic bounds A, B. a. {:i:n} satisfies a lower frame bound of B-1 , i.e., B- 1 11:z:ll 2 ::; :E l(:z:, :i:n)l2 for all :z: EH. b. If {:z:n} is a Bessel sequence with upper bound C then it is a frame, with frame bounds B-1 , C. Moreover, {Yni:Z:n} is in this case a set of atoms, with atomic bounds B-1 , C. PROOF: a. Assume {:i:n; Yn} is a set of atoms. Given :z:, y EH we have 243 , 2 l(x,y)/ = /(x, L(Y,Yn)xn)/2 - /L (x,xn) (Y,Yn)/2 < (L /(x,xn)l 2) (L /(Y,Yn)/ 2) Therefore, llx// 2 = sup /(x,y)/ 2 ~ BL /(x,xn)/2, IIYll=I so {xn} possesses a lower frame bound of B-1 ? b. Assume { Xn} is also a Bessel sequence. Then, by definition, it possesses an upper frame bound. Since it possesses a lower frame bound by part a, {xn} is a frame. It remains to show that {Yni xn} is a set of atoms. The norm equivalence is satisfied since {xn} is a frame, so we need only show that x = L(x, Xn) Yn for all x. Now, both {xn} and {Yn} are Bessel sequences (by assumption for {xn} and by Propostion 6.4.3 for {Yn} ), so by Lemma 6.2.5 the mappings U, V:H--+ l 2 defined by Ux = {(z,xn)} and Vx = {(x,yn)} are linear and continuous, with adjoints U"', V"':l2 --+ H given by U"'{cn} = LCnXn and V"'{ en} = L CnYn? Since {Yni xn} is a set of atoms we have by definition that i.e., U*V = I. Therefore, V"'U = (U"'V)* = r = I, whence x = V*Ux = L (x,xn) Yn? I 244 REMARK 6.4.6. a. In summary, by Remark 6.4.2c all frames are sets of atoms, while by Proposition 6.4.5b all atoms which are also Bessel sequences are frames. By Example 6.4.4, atoms which are not Bessel sequences need not be frames. b. In practice, most sets of atoms are clearly Bessel sequences and therefore are frames. c. Given a set of atoms { Xni Yn} such that { xn} is a Bessel sequence, we have by Proposition 6.4.5 that { xn} is a frame. We also have from Proposition 6.4.3 that {yn} is a frame. However, it need not be true that {Yn} is the dual frame of {xn} or vice versa, as this would imply that atomic coefficients are umque. ,. ,.r,',,. We gave an example of nonunique coefficient functions in Example 6.4.4; .1..,. :., however, that example did not satisfy the Bessel condition. An example in which the Bessel condition is satisfied is the following. Let { en}nEZ+ be an orthonormal basis for H. Then {en, en} is a frame with bounds A= B = 2, and is therefore a Bessel sequence. The dual frame { en/2, en/2} gives one immediate choice for atomic coefficients. However, we can also define atomic coefficients by { en, O}, so they are not unique. d. Nonuniqueness of the atomic coefficients means more than nonunique- ness of the individual representations x = I:(x, Yn} Xn? Nonuniqueness of the individual representations means only that given x there exist some other scalars {en} such that x = L CnXn, Nonuniqueness of the atomic coefficients 245 means that there exists another entire fixed set of vectors {zn} such that x = ~(x, Zn) Xn for all x, moreover, with norm equivalence between 11 ? IIH e. There are a few remarks to be made about the history of Proposition 6.4.5. It was originally believed by Grochenig that all atoms in Hilbert spaces were frames, and he communicated privately a proof of this result to Walnut. We realized that Grochenig's proof implied that the atomic coefficients {yn} are the dual frame of {xn}, and therefore are unique. These results were reported in [Wa], where they are used in a noncritical way for some minor results. Feichtinger later pointed out to us by example that atomic coefficients are not unique. We therefore re-examined Grochenig's proof, and isolated the ??' subtle error. Walnut then gave examples of atoms which were not frames, and suggested the independence of the assumption of the upper frame bound. Finally, we proved Proposition 6.4.5. A special case of Proposition 6.4.5 is proved in [Wa, Theorem 2.6.1]. The following proposition gives a condition under which the atomic coeffi- cient functionals {Yn} will be the dual frame of { xn}, PROPOSITION 6.4.7. Given a set of atoms {xn;Yn} such that {xn} is a Bessel sequence. Define U, V:H---+ ?2 by Ux = {(x,xn)} and Vx = {(x,yn)}. If Range(U) = Range(V) then {yn} is the dual frame of {xn}, PROOF: As is the proof of Proposition 6.4.5 we have U*V = V*U = I, the identity map on H. Let K = Range(U) = Range(V). Since UV*U = UI = 246 U, we have (UV*)IK = IIK- Since Range(U) = Range(V) = K this implies uv?v = IIKV = V. Now, V*V:z: = I: (:z:,yn) Yn = S:z:, wheres is the frame operator for the frame {Yn}? Therefore, given :z: EH, { (:z:, Yn)} V:z: - UV*V:z: - US:z: Since this is true for all z we have Yn = S:z:n, i.e., Zn = s-1Yn? Thus {:z:n} is the dual frame of {Yn}? I f' : I ,!, .(. ,, i.:. ,,. ; .., : .. ? 247 Section 6.5. Frames and atoms in Banach spaces. In this section we extend the notions of frames and atoms to Banach spaces, following the ideas of Grochenig, e.g., [G]. DEFINITION 6.5.1 [G]. Given a Banach space X and a related Banach space Xd of sequences of scalars. Let {xn} be a sequence of elements of X, and let { an} be a sequence of linear functionals on X such that a. X=~an(x)xnforallxEX, b. there exist A, B > 0 such that for all x E X, (.! Then {xn; an} is a set of (Banach) atoms for (X, Xd). A, Bare the atomic ,<' bounds, and { an} are the atomic coefficient functionals. REMARK 6.5.2. a. Often the sequence space Xd will be understood and therefore not specifically mentioned. b. We assume for the remainder of this chapter that each an is continuous, i.e., an EX', and therefore write an(x) = (x,an)? This is true, for example, if Xd is solid and contains each of the sequences {h'rnn}n, for then so arn is continuous on X. c. If {xn} is a basis then {xn} and {an} are biorthonormal, i.e., arn(xn) = 248 DEFINITION 6.5.3. Given a Banach space X and a related Banach space Xd of sequences of scalars. A sequence {Yn} of elements of X' is a (Banach) frame for X if there exist A, B > 0 such that for x E X. A, B are the frame bounds. If only the upper bound holds then the sequence is a (Banach) Bessel sequence. REMARK 6.5.4. a. Definitions 6.5.1 and 6.5.3 for frames and atoms in Banach spaces are consistent with Definitions 6.2.1 and 6.4.1 for frames and atoms in Hilbert spaces. For, the Hilbert space definitions are the special cases of the Banach space definitions obtained by taking X = H and Xd = 1.2 ( except for a square-root factor in the bounds). b. Walnut, in [Wa], discussed the existence of Banach atoms in L~(Rd), where w is a moderate weight. Although L~(Rd) is a Hilbert space, his Banach atoms are not Hilbert atoms since the sequence space is not 1.2 but rather an appropriate weighted?!. Comparing Definitions 6.5.1 and 6.5.3, we obtain the following. PROPOSITION 6.5.5. If {xn; Yn} is a set of atoms for a Banach space X then {Yn} C X' is a Banach frame for X. REMARK 6. 5. 6. Given a set of atoms { x n; Yn} for a Banach space X. As usual, we identify X with its canonical embedding in X", i.e., X C X". Therefore, it possible for { xn} to be a Banach frame for X. The following 249 results give conditions under which this will be true. The situation is similar to the Hilbert space case, in particular, the same considerations about upper and lower frame bounds apply. We assume for the remainder of this chapter that Xd is such that X~ is also a sequence space, with duality between Xd and X~ given by {{bn}, {en}) = :E bncn, This is true, for example, if Xd is a weighted ?P-space. PROPOSITION 6.5.7. Given a set of atoms {:i:niYn} for (X,Xd), with atomic bounds A, B. a. {zn} satisfies a lower frame bound for X of B- 1 ? b. If { :i:n} is a Bessel sequence for X with upper bound C then it is a ~ ??' Banach frame for X', with frame bounds B-1 , C. .,J, ;~J , PROOF: a. Assume {:i:niYn} is a set of atoms. Given :i: EX and y EX' we -~, ~ compute I {:i:, y) I l(~{:i:,yn}Zn, y)I I~ {:i:, Yn} {:i:n, Y}I < 11{{:i:,yn)}llx" 11{{:i:n,Y)}llx~ < B llxllx ll{(zn,Y)}llx~- Therefore, l1Yllx1 - sup l{:i:,y)I ::; Bll{(:i:n,Y)}llx~- llzll=l 250 This establishes the lower frame bound for { z,,}. b. Follows immediately from a. I ..e.: :.,. ,J l'. ~-. .:.~.~ 251 Section 6.6. Stability of atoms. There are many results on the stability of bases in Banach spaces, e.g., [DE; GK; PW; Po]. Typically, these give conditions on the amounts the elements of a basis may be perturbed without affecting the basis property. We formulate in this section an analogous stability theorem for atoms. We continue to assume that X~ is a sequence space of scalars, with duality between xd and x~ given by ({bn},{cn}) = ~bnCn, PROPOSITION 6.6.1. Given a set of atoms { Xni Yn} for (X, Xd), with atomic bounds A, B. Assume Wn E X satisfy R - ll{llxn - Wnll}llx, < s-1 . d Then there exist Zn E X' such that { Wni Zn} is a set of atoms for (X, Xd) with atomic bounds A/(1 + RB), B/(1- RB). Moreover, {wn} is a basis if and only if { Xn} is a basis. PROOF: Given x E X, (6.6.1) < RBllxll? Thus ~ (x, Yn) (xn - wn) converges absolutely in X. Since x = ~ (x, Yn) Xn also converges in X, the series Tx = ~(x,yn) Wn must therefore converge. 252 Clearly T is linear, and, by (6.6.1), III - TII ::; RB < l. Therefore T is invertible, whence for x E X. By definition of atoms, A (6.6.2) < A IIT-1IITll llxllx xllx < II{ (T-1x, Yn) }llxd < B IIT-1xllx < B IIT-111 llxllx- Define the functional Zn E X' by Zn = T- 1Yn, i.e., (x, Zn) = (T-1x, Yn) for x EX. By (6.6.2), so {wn;zn} is a set of atoms for (X,Xd)- Since IITII ::; 11111 + IIT - Ill ::; 1 + RB and 1 1 IIT-111 ::; 1 - II < l-RB' the bounds are as claimed. Finally, assume {xn} is a basis for X. Then {xn} and {Yn} are biorthonor- mal, so 253 Since topological isomorphisms preserve bases, { wn} must therefore be a ba- sis. Conversely, if {wn} is a basis then T-1 is a topological isomorphism which maps {wn} onto {zn}, so {zn} must be a basis. I ,I i',,. ;1 254 CHAPTER 7 GABOR SYSTEMS AND THE ZAK TRANSFORM This chapter is an essentially expository survey of results obtained by using the Zak transform to analyze Gabor systems. We have combined results from (D1; DGM; J2] and others with remarks, examples, and results of our own. In Section 7.1 we define Gabor systems, and give necessary and sufficient conditions under which a Gabor system will be a frame, if the mother wavelet .~J ' " has compact support. .". ?. ,j,,, In Section 7.2 we define the Zak transform and prove that it is a unitary - map of L2 (Rd) onto L 2 (Q), where Q is any unit cube in Rd x Rd. ,I In Section 7.3 we analyze Gabor systems at the critical value ab= l through 1' j , ;i,' the use of the Zak transform. We characterize those systems which are frames by a condition on the Zak transform of the mother wavelet. In Section 7.4 we prove that the Zak transform maps LP(Rd ) into LP( Q) for 1 ::; p :=; 2 but cannot be defined on LP(Rd) if p > 2. In Section 7 .5 we prove that the Zak transform maps the Wiener amalgam space W(LP,L 1 ) into LP(Q) for each 1::; p:::; oo. As a corollary, we obtain a variant of the Balian-Low theorem: if (g,a,b) generates a Gabor frame at the critical value ab = l then g is either not smooth or does not decay quickly at infinity. Finally, in Section 7.6 we address some questions similar to ones which arise 255 in Section 7 .3 from the application of the Zak transform to Gabor frames. In particular, we generalize slightly a result of Boas and Pollard which shows that if finitely many elements are removed from an orthonormal basis for L2 (X) then it is always possible to find a single function to multiply the remaining elements by so that the resulting sequence is complete. We show this need not be true if infinitely many elements are deleted, and discuss some related results by other authors. ,, i ~,,: (. ",~,., ;,I ,I, a~? 256 Section 7.1. Gabor systems. In this section we define Gabor systems in L2 (Rd), and prove an ex1s- tence theorem for Gabor frames generated by compactly supported mother wavelets. DEFINITION 7.1.1. Given g E L2 (Rd) and given a, b E Ri, the Gabor system generated by (g, a, b) is {TnaEmb9}m,nEZ"? The function g is the mother wavelet and the vectors a, b are the system parameters. The set of points { ( na, mb) }m,nEZ" is the system lattice. When a, b are understood we use the abbreviation 9mn = TnaEmb9? REMARK 7.1.2. a. Since TnaEmb = e-2,rinamb EmbTna, we also refer to {EmbTna9}m,nEZ" as a Gabor system. b. Since (TnaEmb9)" = E-naTmb9, the Gabor system generated by (g,b,a) consists of the Fourier transforms of the elements of the Gabor system gener- ated by (g,a,b). Since the Fourier transform is a unitary mapping of L2 (Rd) onto L2 (Rd ) there is a duality between properties held by the system gener- ated by (g, a, b) and the system generated by (g, b, a). c. Assume that the Gabor system generated by (g, a, b) is a frame. The frame operator is then, by definition, Sf= ~(/,9mn) 9mn, and the dual frame is {S-19mn}- A stra.ightforwardcalculationshowsthat STnaEmb = TnaEmbS, whence s-19mn = (s -1 g )mn? Therefore the dual frame is also a Gabor frame, generated by (s- 19,a,b). The following is an elaboration of a basic result from [DGM). 257 PROPOSITION 7.1.3. Given g E L2(Rd) with compact support and given a E Ri. Let I :) supp(g) be any compact rectangle, and let l/b be the side lengths of I. Define A(t) L lg(t - na)l2' A - ess inf A( t), and B - ess sup A( t). tER" nEZ" tER" Then (g, a, b) generates a Gabor frame if and only if A > 0 and B < oo. In this case, the following also hold. a. The bounds for the frame are II(l/b)A, II(l/b)B. b. The frame opera.tor is Sf= II(l/b) Af. c. o ..(t)dt, m,n R" so (g, a, b) forms a frame if and only if).. is essentially constant. Assume now that (g, a, b) generates a frame. a. Follows from (7.1.1). b. For each f E L2(Rd) we have {Sf,f) = ~ l{J,gmn)l2 = JI{l/b) L, lf(t)12 >.(t)dt - JI(l/b) (>.J,J). .l" , , It follows immediately from elementary Hilbert space results that Sf IT(l/b) >..J. l I I c. If ajbj > l for some j then {In}, and therefore {supp(9mn)}, does not I? I? =- cover Rd. Hence {9mn} is incomplete and therefore not a frame. d. Assume ab = l. In this case the sets { In} are disjoint. Therefore A(t) = lg(t - na)l 2 if t E In, whence ITna9I is bounded above and below on In, As {Il(b)112 EmbXr,.}m forms an orthonormal basis for L2 (In), it follows that {EmbTna9}m is a bounded unconditional basis for L2 (Jn), Since {In} is a partition of Rd, it follows that {EmbTna9}m,n is a bounded unconditional basis, and hence an exact frame, for L2(Rd). Conversely, assume ab f:. l. From part c, every coordinate of ab is at most 1. There are two possibilities: either supp(g) #- I or supp(g) n supp(Tka9) f-: 0 for some k E zd. We claim that in either case it is possible to remove one 259 element from the frame {9mn} and retain a complete set, from which it follows that the frame is inexact. In particular, we remove the element g = g00 ? To show that {9mn}cm,n)#(o,o) is complete, assume f E L2(Rd) satisfies (/,9mn) = 0 for (m,n) #- (0,0). Note that supp(!? g) C I and that (/ ?g, Em,b) = (/,9mo) = 0 form-=/:- 0. As {II(b)1l 2 EmbXI}m is an orthonormal basis for L 2 (1), it follows that f ? g = cE0 = c for some constant c. If supp(g) -=f:. I then c = 0 since f ? g = 0 on I\ supp(g ). On the other hand, assume supp(g) n supp(T1:0 g) -=f:. 0 for some k. Then (/ ? Tka9, Emb) = (/,9mk) = 0 for all m, whence f ? l" T1: 0 g = 0 on h, Therefore .. ' f = 0 on supp(Tka9)::) supp(g) n supp(Tka9), so again c = 0. Thus, in any case, f ? g = 0 on I, whence (/,g) = 0 since supp(g) C I. Thus f is orthogonal to every element of {Ymn}? As this set is complete, it follows that f = O, and therefore {9mn}cm,n)#(o,o) is complete. Alternatively, recall from Corollary 6.2.14 that the frame {Ymn} is exact if and only if (9mn, s-19mn) = 1 for all m, n. By part b, (7.1.2) {9mn, s-19mn) = II(b) {9mn,9mn(>..) - II(b) r lg(t - na)l2 dt. Jad I:1: lg(t - ka)l 2 Now, lg(t - na)l 2 (7.1.3) E1: jg(t - ka)j2 ~ X1,.(t). If ab= 1 then there is equality in (7.1.3), and therefore (Ymn, s-19mn) = 1 by (7.1.2). This is true for all m, n, so the frame is exact. 260 Assume, on the other hand, that ab=/=- 1, and set m = n = O. If supp(g) =f=. I then g(t) = 0 fort EE= I\supp(g). If supp(g) n supp(T1:0 g) =/=- 0 for some k E zd then lg(t)l 2 < lg(t)l 2 + lg(t - ka)l 2 fort EE= supp(g) n supp(T1:ag). In either case there is strict inequality in (7.1.3) for t E E. As IEI > O, it follows that (g, s-1 g} < 1, whence {9mn} is inexact. e. Assume ab=/=- l. From part d, {9mn}cm,n)?(o,o) is complete, and therefore is~ frame (Proposition 6.2.12). The argument in part d used only the function g and those 9mn whose support intersected that of g. Therefore the argument ,.. I can be repeated using some 9kl whose support is far distant from that of g and its immediate neighbors, i.e., we can remove some a second function from the frame and still have a complete set and therefore a frame. This process I can be repeated arbitrarily many times, so the frame has infinite excess. I I I [, I' ~ EXAMPLE 7 .1 .4. Functions in Cc(Rd ) satisfy the hypothesis of Proposition 7.1.3 for all a and b whose components are small enough. Therefore, any function in Cc(R d) will generate a Gabor frame for some choice of a and b. It is possible to prove sufficient conditions under which functions without compact support will generate Gabor frames, e.g., [D1; HW; Wa]. 261 Section 7 .2. The Zak transform. In this section we define the Zak transform and prove that it is a unitary mapping of L2 (Rd) onto L2 (Q), where Q is any unit cube in Rd x Rd. DEFINITION 7.2.1. The Zak transform of a function f: Rd -. C is (for- mally) ZJ(t,w) - L J(t+k)E1c(w) l:EZ4 ,, The series defining Z f may converge in various senses, e.g., pointwise, Lf c, 0 etc. Formally, Z f is quasi periodic, in the following sense. DEFINITION 7.2.2. A function F: Rd X Rd-. C is quasiperiodic if F(t + j,w + k) = E_;(w)F(t,w) = e-21rij,w F(t,w) d Ad forj,kEZdand(t,w)ER xR. REMARK 7.2.3. a. A quasiperiodic function is completely determined by its values on any unit cube Qin Rd x Rd. b. If F, G are quasiperiodic then FG is 1-periodic. c. If F is quasiperiodic then the norm is independent of the unit cube Q. Hence {F:Q-. C: IIFll2,Q < oo} 262 can be identified with {F: Rd x Rd-+ C: Fis quasiperiodic and IIFll2,Q < oo} via quasi periodic extension. We refer to both of these spaces as .LP( Q). Without loss of generality, we let Q = [O, 1] x [O, 1] C Rd x Rd for the remainder of this chapter. d. {E(m,n)}m,nEZ,. is an orthonormal basis for the Hilbert space L2( Q), where E (t w) _ e2n(m,n) ?(t,"') _ e2nm?t e2,rin?w (m,n) , - - ? e. Quasiperiodicity is not a translation invariant property. For example, set d = 1 and assume F is quasiperiodic. If b is not an integer then (Tco,r.)F)(t + 1,w) - F(t + 1,w - b) _ e-211-i(w-r.) F(t,w - b) Thus Tco,r.)F is not quasiperiodic. The following proposition and its proof is a generalization to higher dimen- sions of a result appearing in [J2]. PROPOSITION 7.2.4. The Zak transform is a unitary map of L2 (Rd) onto 263 a bWWI Iii S PROOF: Fix f E L2(Rd). Fork E zd define Fk(t,w) = J(t + k)Ek(w). Since IIFkll~ = jrf lf(t+k)Ek(w)l2 dwdt = f IJ(t+k)l2 dt < oo, )q Jro,1) we have Fk E L2 ( Q) for each k. The sequence {FA:} is orthogonal, for if k # l then {Fk, Fi) = f J(t + k) J(t + l) ( f Ek-1(w) dw) dt - 0. Jro,1] Jro,1) Given a finite subset F C zd we therefore have ,, Since L f IJ(t + k)l2 dt = L IJ(t)l2 dt = IIJII~ < oo, kEZ" J[o,l] B." it follows that ZJ = }:Fk converges in L2(Q) and IIZ/112 = 11/112, so Z is continuous and norm-preserving. Now define g = Xro,iJ and set a = b = 1. The Gabor system {Ymn} is then an orthonormal basis for L2(Rd), We easily compute Zgmn = E(m,n)? Thus Z maps the orthonormal basis {Ymn} for L2(Rd) onto the orthonormal basis {E(m,n)} for L2(Q). As Z is continuous, it follows immediately that Z is unitary. I PROPOSITION 7.2.5. Given f E L2 (Rd), Zf(t,w) - e-l,rit?w Z}(w,-t). 264 PROOF: a. Fix cp in the Schwartz space S(Rd); then we can apply the Poisson summation formula in the calculation below, cf., Section 1.8b. Zcp(t,w) - L cp(t + k)Ek(w) - L T_,cp(k) Ew(k) - L (EwT-tcp)(k) L(EwT-tcp)"(k) L ,.. - TwEtcj,(k) - L E,cj,(k -w) - Et(-w) L cj,(k - w) Et(k) - e -2,rit?W z cAp ( - w, t) . b. Now fix / E L2(Rd). Then there exist 'Pn E S(Rd) such that 'Pn -+ f in L2(Rd). By the Plancherel formula, 'Pn -+ j in L2(Rd). By Proposition 7.2.4, it follows that Zcpn-+ Zf and Zcj,n-+ zj in L2(Q). By passing to subsequences if necessary we may assume that all four of these convergences hold pointwise a.e. Therefore, pointwise a.e., Zf(t,w) lim Zcpn(t,w) n-+oo l1o m e -2,rit?w z 'PA n ( -w, t) n-+oo - e-2w-it-w zf(-w,t). I 265 EXAMPLE 7.2.6 [DGM]. Set d = 1. We compute the Zak transform of the Gaussian g(t) = 2e-rt , where r > 0. We make use of the Jacobi theta function 93. There are four Jacobi theta functions, defined by 00 2 L (-lt q sin 21r(2k + l)z ?=o -i L00 (-lt qC?+1/2)2 e2,ri(211+i).c, ?=-co 00 9 (z, q) - 2 L q cos 21r(2k + l)z ?=o L00 (-ll q 0. Since e-r = e'lfi{ir), it follows that Zg(t,w) = 0 if and only if w irt 1 ir m irn 2 + 21r 4 + 47r + 2 + 27r ' i.e., (t,w) = (n + 1/2,m + 1/2). Thus Zg has a single zero in any unit cube 267 Section 7.3. Gabor systems and the Zak transform. In this section we use the Zak transform to analyze Gabor systems satisfying ab= 1. REMARK 7.3.1. Let Da. denote the dilation operator which is isometric on L2 (Rd), i.e., Da.f(t) = III(a)l-1 / 2 f(t/a), and assume ab= 1. Then Since dilation is a unitary operator on L2 (Rd) it therefore suffices to consider Gabor systems satisfying a = b = 1. We assume these values for the remainder of this chapter, i.e., 9mn = TnEm9? LEMMA 7.3.2. Given a function g on Rd and a= b = 1, Zgmn - Ecm,n) Zg. PROOF: We compute Zgmn(t,w) - L 9mn(t + k)E1c(w) - L TnEmg(t + k)E1c(w) - L Em(t + k - n)g(t + k - n)E1c(w) Em(t) L g(t + k)E1c+n(w) - Em(t) En(w) Zg(t,w) - Ecm,n)(t,w )Zg(t,w). I 268 REMARK 7.3.3. Since the Zak transform is a unitary map of L2(Rd) onto L2 ( Q), the Gabor system {9mn} will form a frame for L2(Rd) if and only if {Z9mn} forms a frame for L2(Q). By Lemma 7.3.2, Zgmn = E(m,n)Zg. Since {E(m,n)} forms an orthonormal basis for L2 (Q), the requirement that {E(m,n) Zg} be a frame therefore places severe restrictions on Zg, which we examine in the following proposition. The study of Gabor systems satisfying ab = 1 is thus reduced via the Zak transform to the study of the effect of multiplying the elements of a particular orthonormal basis, {E(m,n)}, by a single fixed function, Zg. There are many related questions which have appeared in the literature; we discuss some of these in Section 7.6. Parts a and d and the frame statement of part c of the following proposition have appeared in print several times, e.g., [DGM]. PROPOSITION 7.3.4. Given g E L2(Rd) and a= b = 1. a. {9mn} is complete in L2(Rd ) if and only if Z g =/. 0 a.e. b. {9mn} is minimal and complete in L2(Rd) if and onlyifl/Zg E L2 (Q). c. {9mn} is a frame for L2 (Rd) (with frame bounds A, B) if and only if 0 < A < IZgl 2 < B < oo a.e. In this case the frame is exact. d. {9mn} is an orthonormal basis for L2(Rd) if and only if jZgj = 1 a.e. PROOF: a. Assume that {9mn} is complete in L2(Rd); then {Z9mn} is com- 269 plete in L 2 (Q) by the unitarity of Z. Define Fon Q by 1, = Zg(t,w) = O, F(t,w) { o, Zg(t,w) =/= 0. Then, by Lemma 7.3.2, form, n E zd, (F,Zgmn) = (F,E(m,n)Zg) = (F ? Zg,E(m,n)) = 0. Therefore F = 0 a.e. since { Z 9mn} is complete, whence Z g =/= 0 a.e. Conversely, assume Zg =/= 0 a.e. Assume that F E L2 (Q) is such that (F, Z 9mn) = 0 for all m, n. Then (F ? Z g, E(m,n)) = (F, Z 9mn) = 0 for all m, n, so F ? Zg = 0 a.e. since F ? Zg E L1(Q) and {E(m,n)} is complete in L 1 (Q). As Zg =/= 0 a.e., this implies F = 0 a.e., so {Z9mn} is complete. b. Assume {9mn} is minimal and complete in L2 (Rd); then the same is true of {Z9mn} in L2 (Q). By part a, Zg =/= 0 a.e. By Proposition 6.1.3, the minimality of {Z9mn} implies that there exist functions Fmn E L2 (Q) which are biorthonormal to { Z 9mn}, i.e., (Fm n, Z 9m'n') = imm' inn'. Thus (Fmn ? Zg,E(m',n')) = (Fmn,Z9m 1n1 ) = imm' Onn' = (E(m,n),E(m',n')), Since Fmn?Zg E L 1(Q) and {E(m',n')} is complete in L1(Q), Fmn?Zg = E(m,n) a.e. for all m, n. Thus E(m,n)/Zg = Fmn E L2(Q) for all m, n; in particular, 1/Zg E L 2 (Q). Conversely, assume 1/Zg E L2 (Q). Then Zg =/= 0 a.e., so {9mn} is complete by part a. Let g = z-1(1/Zg) E L2 (Rd). Then (9mn,9m'n') = (Zgmn, Zgm'n') 270 - (E(m,n), E(m',n')} Thus {9mn} and {?mn} are biorthonormal. The existence of a biorthonormal sequence implies by Proposition 6.1.3 that {9mn} is minimal. c. Assume {9mn} is a frame for L2 (Rd) with frame bounds A, B; then the same is true of { Z 9mn} in L2( Q). Therefore, by definition, for FE L 2 (Q). Since {E(m,n)} is an orthonormal basis for L 2(Q), It follows immediately that A :5 inf jZgj 2 and B 2=: sup jZgj 2 ? Conversely, assume Zg is essentially constant, i.e., A :5 jZgj2 :5 B a.e. Then the mapping U F = F ? Zg is a topological isomorphism of L2(Q) onto itself. Since {E(m,n)} is an exact frame for L2 ( Q) and exact frames are preserved by topological isomorphisms (Lemma 6.3.2), it follows that {U E(m,n)} = {Zgmn} is an exact frame for L2 (Q), whence {9mn} is an exact d. Follows immediately from c. I EXAMPLE 7.3.5. Set d = 1. In Example 7.2.6 we determined that the Zak 2 transform of the Gaussian g( t) = e-rt is continuous and has a zero. Therefore IZ gj is not bounded below a.e., so by Proposition 7.3.4 and Remark 7.3.1, 271 {9rnn} is not a frame for L2(Rd) when ab= 1. However, as Zg is nonzero a.e., this Gabor system is complete. The question of the completeness of this Gabor system was Zak's original motivation for introducing the Zak transform. We prove now that Z g must have a zero if it is continuous. Therefore, by Proposition 7.3.3, no function with a continuous Zak transform can generate a Gabor frame at the critical value ab = 1. By saying Zg is continuous we mean that Zg is continuous on all of Rd x ftd, not just inside a unit cube Q. Equivalently, we require continuity both inside and at the edges of Q. The proof of the following proposition is adapted from the one-dimensional version found in [J2]. The first published proofs were [AT2; BZ]. PROPOSITION 7.3.6. Every continuous quasiperiodic function has a zero. Precisely, given a continuous quasiperiodic function F, fix any j = 1, ... , d, TE Rd, and OE ftd and define T(t) (T1, ... ,Ti-1,t,Ti+1,, .. ,Td), Then there exist t E Rand w ER such that F(T(t), il(w )) = 0. PROOF: Assume F is continuous, quasiperiodic, and nonvanishing. Then f(t,w) = F(T(t),il(w)) is continuous and nonvanishing on Rx ft, so by [RR, Lemma Vl.1. 7] there is a continuous real-valued function '{) such that f (t , w) = If( t, w) I eic,o(t,w) for (t , w) E [0, 1] x [0, 1 ]. It follows immediately from 272 the quasiperiodicity of F that f(t, 1) = f(t, 0) and f(l,w) - e-21riwf(O,w). Therefore, for t, w E (0, 1], lf(t,l)jeiip(t,I) - f(t,1) - f(t,0) lf(t,0)I eiip(t,o) - lf(t,l)leirp(t,o) and lf(l,w)jeirp(l,w) - f(l,w) - e-271'iw f(0,w) - e-271'iw 1/(0,w )I eiip(O,w) - e-271'iw 1/(1,w)leicp(O,w). As f is nonvanishing, it follows that and Therefore, for each t, w E [0, 1] there exist integers kt and lw such that <,0(t, 1) = <,0(t, 0) + 21rkt and 2. PROPOSITION 7.4.1. The Zak transform is a linear, continuous, injective mapping of L 1(R11 ) into L1(Q) with IIZII = 1. Moreover, Range(Z) is dense in L 1( Q) but Z is not surjective, and z-1: Range(Z) _. L1(R11) is not con- tinuous. PROOF: Fix f E L 1(R11 ). Fork E zc1 define Fk(t,w) = f(t+k)Ek(w). Since IIFA:111 = !~J rq IFk(t,w)I dw dt = r lf(t + k)I dt < oo, lro,11 Fk E L1{Q). Moreover, ~ IIFA:lli = ll!ll1, so Zf converges absolutely and JJZJll1 < ~ IIFA:111 = IIJll1, Thus Z is continuous and IIZII 5 1. If g = X[o,l] then Zg,nn = E(m,n), SO IIZII = 1 since IIZBmnl11 = IIE{m,n)lli = 1 = !IBmnlll? Also, Range(Z) is dense since {E{m,n)} is complete in L1(Q). For a.e. t E Rd, r z f(t,w) CUA) f Lf(t + k)Ek(w) L2(Q) = Z(L2 (R')), which implies L1 (R') :> L2(R'), a contradiction. Therefore Z is not surjective. Finally, we show z-1 is not continuous. Given R > 0 there exists a bounded, I-periodic function g E L00 [0, 1) such that ll9lloo ::; 1 and R ~ E l?(k)I < oo, e.g., [K, p. 99J, where {g(k)} are the Fourier coefficients of g, i.e., g(k) = ~o,i)g(t)E-?(t)dt fork E zd. Define f = Eb(k)X[l,l+l]i then 11/111 = E jg(k )I < oo. Since g has an absolutely convergent Fourier series, Zf(t,w) = Lf(t + k)E.(w) = Lg(k)E.(w) = g(w) a.e. Thus IIZ /Iii = IIYll1 ~ ll9lloo ~ 1 while II/Iii > R. As R is arbitrary, this implies z-1 is unbounded and therefore not continuous. I In the course of the proof of Proposition 7 .4.1 we proved the following, cf., [.J2]. COROLLARY 7.4.2. Hf E L1(R') then f(t) = f Zf(t,w)dw J[o,1] for a.e. t ER'. COROLLARY 7.4.3. H / E L1(R') and Zf is continuous then f is continuous. 276 PROOF: If Z f is continuous then it is uniformly continuous on Q, so lf(t)-f(s)I < r IZJ(t,w)-Zf(s,w)ldw J[o,1] < sup IZ f(t,w) - Z f(s,w )I "'E[D,1] -+ 0 as s -+ t. I EXAMPLE 7.4.4. The converse of Corollary 7.4.3 does not hold. To see this, set d = 1 and let

L2 (Q) = Z(L2 (Rd)), whence LP(Rd) :::> L2(Rd), a contradiction. Therefore Z is not surjective, and hence z-1 is not continuous by Lemma 7.4.5. I Since {E(m,n)} is not a basis for L 1 (Q), the method of Proposition 7.4.6 cannot directly be used to prove Proposition 7.4.1. EXAMPLE 7.4.7. The Zak transform cannot be defined as a map of Lq(Rd) onto any Lr( Q) when q > 2. For example, by [K, p. 100], there exist scalars {en} such that I: JCA1lq < oo for every q > 2 but I: c1cE1c is not a Fourier series. Define f = I:c1cX[1c,1c+1]; then/ E Lq(Rd) but Zf(t,w) = I:f(t+k)E1c(w) = I:c1cE1:(w) does not converge in any Lr(Q). 279 Section 7.5. Amalgam spaces and the Zak transform. In this section we examine the convergence of the Zak transform on Wiener amalgam spaces, in particular, on the amalgam space W (LP, L 1 ) on the ad- ditive group Rd. We prove that the Zak transform maps W(LP,L1 ) into LP( Q) for each 1 ~ p ~ oo, and maps W( Co, L1 ) into the space of continuous quasiperiodic functions. This gives us a variant of the Balian-Low theorem, i.e., if (g, a, b) generates a Gabor frame at the critical value ab = 1 then g ?_ W(C ,L10 ), whence g is either not continuous or decays slowly at infinity. From Example 2.4.4, the Wiener amalgam space W(LP,L 1 ) is defined by the norm We adopt this as the standard norm for W(LP, L 1 ). Also, from Example 2.2.4, W(Co,L 1 00 1) = {f E W(L ,L ): f is continuous}, with the W(L 00 L 1, ) norm. PROPOSITION 7.5.1. Given 1 ~ p::; oo, the Zak transform is a continuous, linear, injective map ofW(LP,L1 ) into LP(Q), with IIZII = 1. PROOF: Fix f E W(LP,L1 ). Fork E zd define Fk(t,w) = f(t + k)Ek(w). Then Fk E LP(Q) since IIFA:IIP = llf ? X[k,k+1JIIP < oo. Moreover, L IIFkllp = L llf ? X[k,k+i]IIP = llfllw(v?,L1 ) < oo, so Zf = EFk converges absolutely in W(LP,L1 ), and IIZII ~ 1. We have IIZII = 1 as IIZ9mnllP = IIE(m,n)IIP = 1 = l!Bmnllw(L,,L1), where g = X[o,1]? 280 Z is injective as W(D',L1 1) C L (R') and Z is injective on L1(Rci) (Propo- sition 7.4.1 ). I COROLLARY 7.5.2. If f E W(Co,L1 ) then Zf is continuous on Rei X ftci. PROOF: If/ E W(Co,L1 ) then the series defining Zf converges in L00(Q), i.e., uniformly on Q, by Proposition 7.5.1, since W(C0 ,L1 ) C W(L 00 ,L1 ). As each term f(t + k)E1,(w) in the series defining Zf is continuous, it follows that the series must converge to a function which is continuous on Q, and therefore, by quasiperiodicity, on all of Rei X ftci. 1 Corollary 7 .5.2 can a.lso be proved by noting that translation and modula- tion are both strongly continuous in W(C0 ,L1 ). The following is a variant of the Balian-Low theorem. COROLLARY 7.5.3. Given g E L2(Rci) and a= b = 1. If (g,a,b) generates a Gabor frame for L2 (Rci) then g, g ~ W(C0 ,L1 ). PROOF: If g E W(Co, L1 ) then Zg is continuous by Corollary 7.5.2. By Proposition 7.3.5, Zg therefore has a zero, so IZgl is not bounded below a.e. Therefore (g, a, b) cannot generate a frame (Proposition 7.3.3c). Similarly, if g E W(C 10 ,L ) then (g,b,a) cannot generate a frame for L2 (Rci), and therefore, by Remark 7.1.2b, (g,a,b) cannot generate a frame for L2 (Rc1). I REMARK 7.5.4. The usual Balian-Low theorem states that if (g,a,b) gener- ates a Gabor frame for L2(R) and ab = 1 then lltg(t)ll2 ll-r?(,)ll2 = oo, cf., 281 [Bal; Bat; BHW; D1; DJ; Low]. EXAMPLE 7.5.5. The Gaussian function g(t) = e-rt?t, t E Ref, is an element of W(C0 ,L1 ), therefore does not generate a Gabor frame a.t the critical value ab= 1. We proved this directly (ford= 1) in Example 7.3.4, EXAMPLE 7.5.6. Note that W(Co,L1 1) c 0 0(Rcf)nL (Rc1). Although the Zak transform of any element of W( Co, L1 ) is continuous by Corollary 7.5.3, there exist elements of O0 (Rcf) n L1(Rcf) whose Zak transform is not continuous. We constructed an example in Example 7 .4.4. 282 Section 1.6. Multiplicative completion. In this section we address some questions similar to ones which arose during our study of Gabor frames and the Zak transform, cf., Remark 7.3.3. Our motivation is the following question asked by Boas and Pollard, e.g., [BPo). Given an incomplete sequence {fn} of functions in L2(a,b), where (a,b) is an interval in R, when is it possible or impossible to find a function m such that {m ? fn} is complete in L2(a, b)? They proved that if {fn} is obtained by deleting finitely many elements from an orthonormal basis for L2(a,b) then it is always possible to find such a function m, while for the orthonormal sequence {E2n}nez in L2 (0, 1) it is impossible to find such a function m. We elaborate on these two results, then comment on related work by other authors. In this section we use the following definition of &olid, which differs slightly from the one in Section 1.7c. Given a measure space (X,?), a Banach space A of functions on X is solid if g E A and IfI :5 19 I a.e. implies f E A and IIJIIA :5 ll9IIA? LEMMA 7.6.1. Given a measure space (X,?) and given a solid Banach space A of functions on X. Assume that for any set E C X there exists a functfon .,;, on X such that supp(.,;,) C E, 'Ip is finite a.e., and 'Ip ? A. Then, given any Ji, ... , f N E A there exists a function g E L00(X) with g -:/: 0 a.e. such that f /g fl_ A if f E span{f1,??? ,JN} \ {O}. PROOF: Without loss of generality we assume fn -:/: 0 for all n. We proceed 283 by induction. a. Set N = I, and assume / = c/i, where c -=I= 0. As f -=I= 0, there exists a set E C X with positive measure such that lf(t)I 2: e > 0 for a.e. t E E. By hypothesis there then exists a function VJ ~ A with supp( 1P) C E which is finite a.e. Set ip(t) = max{l?(t)1, I}, and let g = I/ip. Then g =5 I a.e. as ip > I a.e., and g -=I= 0 a.e. as ip is finite a.e. Moreover, if t E E then lf(t)/g(t)I 2: eip(t) 2: e 1?(t)/. This also holds fort~ E since supp(?) CE. As VJ ~ A and A is solid, it follows that / / g ~ A. b. Assume now that the conclusion of the lemma holds for some N 2: I, and let /1, ... , /N+I E A be fixed. Then, by hypothesis, there exists a function g E L 00(X) with g # 0 a.e. such that SN - {f E span{/i, ... ,/N} \ {0}: / /g EA} - 0. Define SN+l = {/Espan{fi, .. ,,/N+1}\{o}://gEA}. If SN+1 = 0 then the proof is complete, so assume F = Ef +l cnfn E SN+1? Note that CN+l -=I= 0, for otherwise F E SN, Assume also that G = I:f +l bnfn E SN+1i then bN+1 -=I= 0 for the same reason. Clearly, H - - 1-F - 1 -bG E span{fi, ... ,/N}, CN+l N+l Moreover, H/g E A as both F/g, G/g E A. As SN = 0, it follows that H = 0. Thus G is a multiple of F, so SN C {cF: c -=I= 0}. Now, F -=I= 0 since 284 F E SN+I? Therefore there exists a set E C X with positive measure such that IF(t)I ~ e > o for a.e. t E E. By hypothesis there then exists a function 'IP ff. A with supp(?) c E which is finite a.e. Set cp(t) = max{l?(t)j,1/jg(t)j} and define h = 1/cp. Then h is finite a.e. since g I O a.e., and h ~ 191 so h E L 00{X). Moreover, if t E E then IF(t)/h(t)I 2: e cp(t) ~ e 11/J(t)j. This also holds for t ff. E since supp( 'If') C E. As 'If' (/. A and A is solid, it follows that F/h (/. A. Finally, to finish the proof, assume that f E span{/1, ... , /N+1} \ {O} is given. If f /h EA then f Jg EA since h ~ 191? Therefore f E SN+t, so f = cF for some c 'I 0. However, F / h ~ A, a contradiction. Therefore f / h ~ A, so the result follows. I PROPOSITION 7.6.2. Given a measure space (X,?) and a solid Banach space B of functions on X. Assume that B' is also a. solid Banach function space on X which satisfies the hypotheses of Lemma. 7.6.1. Given SC B, defi.ne 5.l. = {g EB': (f,g} = 0 for f ES}. Assume {fn}nEZ+ CB and g1 , ??? ,9N EB' satisfy {fn }J. C span{g1,, ?,, 9N }. Then there exists a. function m E L00 (X) with m f= 0 a..e. such that {m. fn} is complete in B. PROOF: By Lemma. 7.6.1 there exists a function m E L 00(X) with m f= O a.e. such that 285 (7.6.1) g E span{g1,???,9N} \ {O} ? g/in ft. B'. Assume h E B' satisfies (m ? in, h) = 0 for all n. Since m E L 00 (X) we have h ? in EB'. Since (in, h ?in)= 0 for all n, h ? in E {in}.1 C span{g1, ... ,9N}. If h ? in =f:. 0 then h = (h ? in)/in ~ B', a contradiction. Therefore h. in= O, which implies h = 0 a.e. as m =/. 0 a.e., so {m ? in} is complete in B by Definition 6.1.lb. I EXAMPLE 7.6.3. a. Assume {in} CB and {gn} CB' satisfy g = E(g, in) 9n for g E B' (not necessarily uniquely), and fix N > 0. If g E {in};>N then 9 = Ef {9,in)9n E span{g1,???,9N}. b. If {in }nEZ+ is a basis for B and B is reflexive, then there exists a dual ba&is {gn}neZ+ C B', i.e., g = "?(g, in) 9n, uniquely, for all g E B', e.g., [S], cf., Remark 6.1.2. Therefore, by part a and Proposition 7.6.2, given any N > 0 there exists a function m E L 00 (X) such that {m ? in}n>N is complete in B. c. If {gn}nEZ+ is a frame for B = L2 (X) and {in}nEZ+ is its dual frame, then g = E(g, in) 9n for all g E L2(X). Therefore, by part a and Proposi- tion 7 .6.2, given any N > 0 there exists a function m E L00(X) such that {m ? in}n>N is complete in L 2 (X). d. Let X be a finite set and let ? be counting measure on X. Given 0 =/.EC X and any finite function VJ on X with supp(VJ) CE, ll1Pll1P(X) - I: IVJ(t)IP < 00 , tEE 286 since X is finite. Thus A = .LP(X) does not satisfy the hypotheses of Lemma 7.6.1. By Remark 7.6.3b, if finitely many elements are removed from a basis ( thereby leaving an incomplete set) then it is possible to :find a single func- tion m to multiply the remaining elements by to obtain a complete set. We now show by example this need not be true if infinitely many elements are removed, cf., [BPo]. We assume d = l in Lemma 7.6.4 and Proposition 7.6.5. Functions in 2 ? (0, 1) are considered to be extended I-periodically to the entire real line. LEMMA 7.6.4. If f E ? 2 (0,1) is l/N-periodic, whereN E Z+, then (!,En)= 0 for al.I n E Z such that N does not divide n. PROOF: If/ is 1/N-periodic then (I, En) LI f(t) e-2,rint dt N-1 /1/N ~ Jo f(t + k/N) e-2,rin(t+lc/N) dt NL-1 11/ N f(t) e-2,rint e-2,rinlc/N dt lc==O O = Jo/1 /N ) (N-1 ( f(t) ,-??int dt ~ ,-2mn>/N) 2 Let = 1 z e- ,rin/N and w = ~{:'- zlc. Then wz = w - l + zN = w. If N does not divide n then z # l, sow= 0 and therefore(/, En) = O. I PROPOSITION 7.6.5. If S C Z contains an arithmetic progression then the 287 ---- __,__ __ ? L"O 1) by multipli- orthononna.J sequence { En}n;s cannot be comp1 e t e d m L ' cation by an integrable function. PROOF: Without loss of generality, assume S = {nN}neZ for some NEZ+. Fix any m E L 1 (0, 1). If the measure of the zero set of mis positive then {m ? Era}rafS is incomplete, so assume m :/: 0 a.e. Then there exiSts a set E1 C (O, 1/N) on which /m/ is hounded above and below, and then a set E2 C E1 + 1/N on which /m/ is bounded above and below, and so forth ' Define Then F+ 1/N = F (mod 1). Moreover, F C ENU? ? -UEi, so /m/ is bounded above and below on F. Therefore, f(t) = { 1/m(t), t E F, o, t (/. F, IS an nonzero element of L 2[o, 1). Further,/. m = XF is 1/N-periodic, so (/, m ? Era) = (/ ?m , En) = 0 for all n (/. S by Lemma 7.6.4. Thus { m ? Era}n;s is incomplete. I REMARK 7.6.6. Given a sequence {/n}neZ+ C L 2(X), where (X,?.) is a finite separable measure space with ?.(X) = 1, Ta1a1yan proved that the following statements are equivalent, e.g., [Ta]. a. Given t > 0 there exists Ss C X such that ?(Ss) > 1-t and {/n ,Xs.} is complete in L 2(Ss)? 288 b. For every function f on X which is finite a.e. and every e > 0 there exists Se C X andg E span{fn} such that ?(Se)> 1-e and 1/-gj < e on Se, Price and Zink proved that a and b are also equivalent to the following, seemingly unrelated, Boas-Pollard property, e.g., [Pr; PZ]. c. There exists a bounded, nonnegative function m such that { m ? /n} is complete in L2 (X). REMARK 7.6.7. In [Byl; By2; BN], Byrnes and Newman consider a prob- lem similar to the one addressed by Boas and Pollard. Instead of deleting ele- ments from a sequence and then multiplying the remaining elements by a func- tion, they retain all elements of the sequence and multiply only a portion of the sequence by a function. 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