A Panorama of Harmonic Analysis

AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 27

Steven G. Krantz A Panorama of Harmonic Analysis Originally published by The Mathematical Association of America, 1999. ISBN: 978-1-4704-5112-7 LCCN: 99-62756

Copyright © 1999, held by the American Mathematical Society Printed in the United States of America. Reprinted by the American Mathematical Society, 2019 The American Mathematical Society retains all rights except those granted to the United States Government. ⃝∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 24 23 22 21 20 19 10.1090/car/027

AMS/MAA THE CARUS MATHEMATICAL MONOGRAPHS

VOL 27

A Panorama of Harmonic Analysis

Stephen G. Krantz THE CARUS MATHEMATICAL MONOGRAPHS

Published by THE MATHEMATICAL ASSOCIATION OF AMERICA

Committee on Publications William Watkins, Chair

Carus Mathematical Monographs Editorial Board Steven G. Krantz, Editor Robert Burckel John B. Conway Giuliana P. Davidoff Gerald B. Folland Leonard Gillman The following Monographs have been published:

1. Calculus of Variations, by G. A. Bliss (out of print) 2. Analytic Functions of a Complex Variable, by D. R. Curtiss (out of print) 3. Mathematical Statistics, by H. L. Rietz (out of print) 4. Projective Geometry, by J. W. Young (out of print) 5. A History of Mathematics in America before 1900, by D. E. Smith and Jekuthiel Ginsburg (out of print) 6. Fourier Series and Orthogonal Polynomials, by Dunham Jackson (out of print) 7. Vectors and Matrices, by C. C. MacDuffee (out of print) 8. Rings and Ideals, by N. H. McCoy (out of print) 9. The Theory of Algebraic Numbers, second edition, by Harry Pollard and Harold G. Diamond 10. The Arithmetic Theory of Quadratic Forms, by B. W. Jones (out of print) 11. Irrational Numbers, by Ivan Niven 12. Statistical Independence in Probability, Analysis and Number Theory, by Mark Kac 13. A Primer of Real Functions, third edition, by Ralph P. Boas, Jr. 14. Combinatorial Mathematics, by Herbert J. Ryser 15. Noncommutative Rings, by I. N. Herstein (out of print) 16. Dedekind Sums, by Hans Rademacher and Emil Grosswald 17. The Schwarz Function and its Applications, by Philip J. Davis 18. Celestial Mechanics, by Harry Pollard 19. Field Theory and its Classical Problems, by Charles Robert Hadlock 20. The Generalized Riemann Integral, by Robert M. McLeod 21. From Error-Correcting Codes through Sphere Packings to Simple Groups, by Thomas M. Thompson 22. Random Walks and Electric Networks, by Peter G. Doyle and J. Laurie Snell 23. Complex Analysis: The Geometric Viewpoint, by Steven G. Krantz 24. Knot Theory, by Charles Livingston 25. Algebra and Tiling: Homomorphisms in the Service of Geometry, by Sherman Stein and Sandor´ Szabo´ 26. The Sensual (Quadratic) Form, by John H. Conway assisted by Francis Y. C. Fung 27. A Panorama of Harmonic Analysis, by Steven G. Krantz

In homage to Abram Samoilovitch Besicovitch (1891–1970), an extraordinary geometric analyst.

Contents

Preface xiii 0 Overview of Measure Theory and Functional Analysis 1 0.1 Pre-Basics ...... 1 0.2 A Whirlwind Review of Measure Theory ...... 2 0.3 The Elements of Banach Space Theory ...... 13 0.4 Hilbert Space ...... 21 0.5 Two Fundamental Principles of Functional Analysis . . . . 26 1 Fourier Series Basics 31 1.0 The Pre-History of Fourier Analysis...... 31 1.1 The Rudiments of Fourier Series ...... 42 1.2 Summability of Fourier Series ...... 49 1.3 A Quick Introduction to Summability Methods ...... 55 1.4 Key Properties of Summability Kernels...... 62 1.5 Pointwise Convergence for Fourier Series...... 71 1.6 Norm Convergence of Partial Sums and the Hilbert Transform ...... 76 2 The Fourier Transform 95 2.1 Basic Properties of the Fourier Transform...... 95 2.2 Invariance and Symmetry Properties of the Fourier Transform ...... 98 ix x Contents

2.3 Convolution and Fourier Inversion ...... 103 2.4 The Uncertainty Principle ...... 117 3 Multiple Fourier Series 121 3.1 Various Methods of Partial Summation ...... 121 3.2 Examples of Different Types of Summation ...... 126 3.3 Fourier Multipliers and the Summation of Series ...... 130 3.4 Applications of the Fourier Multiplier Theorems to Summation of Multiple Trigonometric Series ...... 140 3.5 The Multiplier Problem for the Ball ...... 148 4 Spherical Harmonics 171 4.1 A New Look at Fourier Analysis in the Plane ...... 171 4.2 Further Results on Spherical Harmonics ...... 183 5 Fractional Integrals, Singular Integrals, and Hardy Spaces 199 5.1 Fractional Integrals and Other Elementary Operators. . . . 199 5.2 Prolegomena to Singular Integral Theory ...... 203 5.3 An Aside on Integral Operators ...... 208 5.4 A Look at Hardy Spaces in the Complex Plane ...... 209 5.5 The Real-Variable Theory of Hardy Spaces ...... 219 5.6 The Maximal-Function Characterization of Hardy Spaces ...... 225 5.7 The Atomic Theory of Hardy Spaces ...... 227 5.8 Ode to BMO...... 229 6 Modern Theories of Integral Operators 235 6.1 Spaces of Homogeneous Type ...... 235 6.2 Integral Operators on a Space of Homogeneous Type . . . 241 6.3 A New Look at Hardy Spaces ...... 262 6.4 The T (1) Theorem ...... 266 7 Wavelets 273 7.1 Localization in the Time and Space Variables ...... 273 7.2 Building a Custom Fourier Analysis...... 276 Contents xi

7.3 The Haar Basis ...... 279 7.4 Some Illustrative Examples ...... 285 7.5 Construction of a Wavelet Basis ...... 297 8 A Retrospective 313 8.1 Fourier Analysis: An Historical Overview ...... 313 Appendices and Ancillary Material 315 Appendix I, The Existence of Testing Functions and Their Density in L p ...... 315 Appendix II, Schwartz Functions and the Fourier Transform . . 317 Appendix III, The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin ...... 318 Appendix IV, Hausdorff Measure and Surface Measure...... 320 Appendix V, Green’s Theorem ...... 323 Appendix VI, The Banach-Alaoglu Theorem ...... 323 Appendix VII, Expressing an Integral in Terms of the Distribution Function ...... 324 Appendix VIII, The Stone-Weierstrass Theorem ...... 324 Appendix IX, Landau’s O and o Notation ...... 325 Table of Notation 327 Bibliography 339

Index 347

Preface

The history of modern harmonic analysis dates back to eighteenth cen- tury studies of the wave equation. Explicit solutions of the problem

∂2u ∂2u − = 0 ∂x2 ∂t2 u(x, 0) = sin jx or cos jx were constructed by separation of variables. The question naturally arose whether an arbitrary initial data function f (x) could be realized as a superposition of functions sin jx and cos jx. And thus Fourier analysis was born. Indeed it was Fourier [FOU] who, in 1821, gave an explicit means for calculating the coefﬁcients a j and b j in a formal expansion X X f (x) ∼ a j cos jx + b j sin jx. j j

The succeeding 150 years saw a blossoming of Fourier analysis into a powerful set of tools in applied partial differential equations, mathematical physics, engineering, and pure mathematics. Fourier analysis in the noncompact setting—the Fourier transform—was developed, and the Poisson summation formula was used, to pass back and forth between Fourier series and the Fourier transform. xiii xiv Preface

The 1930s and 1940s were a relatively quiet time for Fourier analysis but, beginning in the 1950s, the focus of Fourier analysis became singular integrals. To wit, it rapidly developed that, just as the Hilbert transform is the heart of the matter in the study of Fourier series of one variable, so singular integrals usually lie at the heart of any nontrivial problem of several-variable linear harmonic analysis. The Calderon-´ Zygmund theory of singular integrals blossomed into the Fefferman- Stein-Weiss theory of Hardy spaces; Hardy spaces became the focus of Fourier analysis. In the 1980s, two seminal events served to refocus Fourier analysis. One was the David-Journe-Semmes´ T (1) theorem on the L2 boundedness of (not necessarily translation-invariant) singular integrals. Thus an entirely new perspective was gained on which types of singular integrals could induce bounded operators. Calderon´ commutators, Hankel operators, and other classical objects were easy pickings using the powerful new tools provided by the T (1) theorem, and more generally the T (b) theorem. The other major event of the 1980s was the development of wavelets by Yves Meyer in 1985. Like any good idea, wavelet theory has caused us to “reinvent” Fourier analysis. Now we are no longer bound to model every problem on sine waves and cosine waves. In- stead, we can invent a Fourier analysis to suit any given problem. We have powerful techniques for localizing the problem both in the space variable and the phase variable. Signal processing, image compression, and many other areas of applied mathematics have been revolutionized because of wavelet theory. The purpose of the present book is to give the uninitiated reader an historical overview of the subject of Fourier analysis as we have just described it. While this book is considerably more polished than merely a set of lectures, it will use several devices of the lecture: to prove a theorem by considering just an example; to explain an idea by considering only a special case; to strive for clarity by not stating the optimal form of a theorem. We shall not attempt to explore the more modern theory of Fourier analysis of locally compact abelian groups (i.e., the theory of group Preface xv characters), nor shall we consider the Fourier analysis of non-abelian groups (i.e., the theory of group representations). Instead, we shall re- strict attention to the classical Fourier analysis of Euclidean space. Prerequisites are few. We begin with a quick and dirty treatment of the needed measure theory and functional analysis. The rest of the book uses only elementary ideas from undergraduate real analysis. When a sophisticated idea is needed, it is quickly introduced in context. It is hoped that the reader of this book will be imbued with a sense of how the subject of Fourier analysis has developed and where it is heading. He will gain a feeling for the techniques that are involved and the applications of the ideas. Even those with primary interests in other parts of mathematics should come away with a knowledge of which parts of Fourier analysis may be useful in their discipline, and also where to turn for future reading. In this book we indulge in the custom, now quite common in harmonic analysis and partial differential equations, of using the same let- ters (often C or C0 or K ) to denote different constants—even from line to line in the same proof. The reader unfamiliar with this custom may experience momentary discomfort, but will soon realize that the prac- tice streamlines proofs and increases understanding. It is a pleasure to thank the many friends and colleagues who have read and commented on portions of various preliminary drafts of this book. I mention particularly Lynn Apfel, Brian Blank, John Mc- Carthy, Dylan Retsek, Richard Rochberg, Mitchell Taibleson, Guido Weiss, and Steven Zemyan. J. Marshall Ash and Victor Shapiro gave me expert help with the history and substance of the theory of multiple Fourier series. And now for some special thanks: Robert Burckel aimed his sharp and critical eye at both my English and my mathematics; the result is a greatly improved manuscript. Jim Walker contributed many incisive remarks, particularly stemming from his expertise in wavelets; in ad- dition, Walker provided ﬁgures (that were generated with his software FAWAV)for Chapter 7. Gerald B. Folland kept me honest and, acting as Chair of the Carus Committee, helped me to craft the mathematics and xvi Preface to ensure that this project came out as it should. The assistance provided by these three scholars has been so extraordinary, and so extensive, that I sometimes feel as though this manuscript has four authors. Of course responsibility for the extant manuscript lies entirely with me. I am always happy to receive reports of errors or suggestions for improvement.

Steven G. Krantz St. Louis, Missouri Appendices and Ancillary Material

Appendix I The Existence of Testing Functions and Their Density in L p

Consider functions on R1. The real function

( 2 e−1/x if x > 0 φ(x) = 0 if x ≤ 0 is inﬁnitely differentiable (C∞) on the real line (use l’Hopital’sˆ Rule). As a result, the function

ψ(x) = φ(x + 1) · φ(−x + 1) is C∞ and is identically zero outside the set (−1, 1). We say that ψ is a “C∞ function with compact support.” The support of ψ—that is, the closure of the set on which ψ is nonzero—is the interval [−1, 1]. Given any compact interval [a, b], it is clear that this construction may be adapted to produce a C∞ function whose support is precisely [a, b]. ∞ ∞ The C functions with compact support (usually denoted Cc or ∞ p Cc (R) for speciﬁcity) form a dense subset of L (R), 1 ≤ p < ∞. To see this, ﬁrst note that the characteristic function of an interval I p ∞ may be approximated in L norm by a Cc function (see Figure 1). We 315 316 Appendices and Ancillary Material

∞ Figure 1. Approximating a characteristic function by a Cc function.

−1 achieve that approximation by considering ψ(x) ≡ ψ(x/) and then letting u = ψ ∗ χI . Next, we note that by construction the simple functions are dense in L p. By the outer regularity of measure, any simple function in L p may be approximated by a ﬁnite sum of characteristic functions of intervals. Each characteristic function of an interval may be approximated in L p norm by a smooth “bump,” as in the last paragraph. That completes a sketch of the proof. Finally, we use this information to prove that, if 1 ≤ p < ∞ and f ∈ L p, then

lim k f − τa f kL p = 0. a→0

The assertion is plainly true if f is continuous and compactly supported p (i.e., f ∈ Cc), hence uniformly continuous. For any L function f , let > 0 and choose φ ∈ Cc such that k f − φkL p < . Then

k f − τa f kL p ≤ k f − φkL p + kφ − τaφkL p + kτaφ − τa f kL p

= k f − φkL p + kφ − τaφkL p + kτa(φ − f )kL p Appendices and Ancillary Material 317

≤ + kφ − τaφkL p + kφ − f kL p

≤ + kφ − τaφkL p + .

The middle expression tends to zero because φ is uniformly continuous (and has compact support). So the result is proved. We conclude this Appendix by making several remarks. First, all of the constructions presented so far work in higher dimensions, with only small modiﬁcations. Second, on the circle group T, the trigonometric polynomials also form a useful dense set in L p, 1 ≤ p < ∞. This can be seen by imitating the constructions already given, or by using the Stone-Weierstrass theorem (Appendix VIII). Lastly, if (as in the proof of Proposition 2.1.2), one has a differentiable f ∈ L1 with 1 derivative ∂ f/∂x j in L , then slight modiﬁcations of our arguments in ∞ this Appendix allow one to approximate such an f by Cc functions φk 1 1 so that φk → f in the L topology and ∂φk/∂x j → ∂ f/∂x j in the L topology. Variants of this last remark will be used throughout the book.

Appendix II Schwartz Functions and the Fourier Transform

A Schwartz function φ is an inﬁnitely differentiable function such that, for any multi-indices α and β, the expression

∂ ≡ α ρα,β (φ) sup x β φ(x) x ∂x is ﬁnite. We let S denote the space of Schwartz functions, and topolo- 0 gize it using the semi-norms ρα,β . Because of the facts that df (ξ) = 0 −iξ bf (ξ) and [ bf ] (ξ) = ixd f (ξ), it is easy to see that the Fourier transform takes Schwartz functions to Schwartz functions. So does the inverse Fourier transform. Since the Fourier transform is one-to-one, it follows thatbis a bicontinuous isomorphism of S to itself. 318 Appendices and Ancillary Material

Appendix III The Interpolation Theorems of Marcinkiewicz and Riesz-Thorin

The simplest example of an interpolation question is as follows. Sup- pose that the linear operator T is bounded on L1 and bounded on L2. Does it follow that T is bounded on L p for 1 < p < 2? [The space L p here is an instance of what is sometimes called an “intermediate space” between L1 and L2.] Note that this question is similar to (but not precisely the same as) one that we faced when considering the L p boundedness of Calderon-Zygmund´ singular integral operators. Here we record (special) versions of the Riesz-Thorin Theorem (epitomizing the complex method of interpolation) and the Marcinkiewicz Interpo- lation Theorem (epitomizing the real method of interpolation) that are adequate for the applications in the present book.

Theorem (Riesz-Thorin): Let 1 ≤ p0 < p1 ≤ ∞. Let T be a linear operator on L p0 ∩ L p1 such that

kT f kL p0 ≤ C0 · k f kL p0 and

kT f kL p1 ≤ C1 · k f kL p1 . If 0 ≤ t ≤ 1 and 1 1 1 − t t = = + , p pt p0 p1 then we have

1−t t k k p ≤ · · k k p T f L C0 C1 f L .

Recall that, for 1 ≤ p < ∞, we say that a measurable function f is weak-type p if there is a constant C > 0 such that, for every λ > 0, C m{x : | f (x)| > λ} ≤ . λp Appendices and Ancillary Material 319

We say that f is weak-type ∞ if it is just L∞. A linear operator T is said to be of weak-type (p, p) if there is a constant C > 0 such that, for each f ∈ L p and each λ > 0,

p k f k p m{x : |T f (x)| > λ} ≤ C · L . λp An operator is weak-type ∞ if it is simply bounded on L∞ in the classical sense. An operator is said to be strong-type (p, p) (or, more generally, (p, q)) if it is bounded from L p to L p (or from L p to Lq ) in the classical sense discussed in Chapter 0. Now we have

Theorem (Marcinkiewicz): Let 1 ≤ p0 < p1 ≤ ∞. If T is a (sub-) p p linear operator on L 0 ∩ L 1 such that T is of weak-type (p0, p0) and also of weak-type (p1, p1), then for every p0 < p < p1 we have

kT f kL p ≤ C p · k f kL p .

Here the constant C p depends on p and, in general, will blow up as → + → − either p p0 or p p1 .

These interpolation theorems have been generalized in a number of respects. If T is a (sub-)linear operator that maps L p0 to Lq0 and L p1 q to L 1 (either weakly or strongly), and if p0 < p1, p0 ≤ q0, p1 ≤ q1, then T maps L pt to Lqt strongly, where

1 1 1 = (1 − t) · + t · pt p0 p1 and 1 1 1 = (1 − t) · + t · . qt q0 q1 We shall occasionally use this more general version of Marcinkiewicz interpolation. 320 Appendices and Ancillary Material

Appendix IV Hausdorff Measure and Surface Measure

If ⊆ RN has C1 boundary, then we use the symbol dσ to denote (N − 1)-dimensional area measure on ∂. This concept is fundamental; we discuss, but do not prove, the equivalence of several definitions for dσ. A thorough consideration of geometric measures on lower-dimensional sets may be found in the two masterpieces [FE1] and [WHI]. First we consider a version of a construction due to Hausdorff. Let N S ⊆ R and δ > 0. Let U = {Uα}α∈A be an open covering of S. Call U a δ-admissible covering if each Uα is an open Euclidean N-ball of radius 0 < rα < δ. If 0 ≤ k ∈ Z, let Mk be the usual k-dimensional k Lebesgue measure of the unit ball in R (e.g., M1 = 2, M2 = π, M3 = 4π/3, etc.). Deﬁne

( k X k Hδ (S) = inf Mkrα : U = {Uα}α∈A α∈A is a δ-admissible cover of S} .

k ≤ k 0 k Clearly, Hδ (S) Hδ0 (S) if 0 < δ < δ. Therefore limδ→0 Hδ (S) exists in the extended real number system. The limit is called the k- dimensional Hausdorff measure of S and is denoted by Hk(S). The function Hk is an outer measure.

Exercises for the Reader IV.1. If I ⊆ RN is a line segment, then H1(I ) is the usual Euclidean length of I. Also, H0(I ) = ∞ and Hk(I ) = 0 for all k > 1. IV.2. If S ⊆ RN is Borel, then HN (S) = LN (S), where LN is Lebesgue N-dimensional measure. IV.3. If S ⊆ RN is a discrete set, then H0(S) is the number of elements of S. Appendices and Ancillary Material 321

α IV.4. Define Mα = 0(1/2) / 0(1 + α/2), and α > 0 (note that this is α consistent with the preceding definition of Mk). Then define H for any α > 0 by using the Hausdorff construction. Let S be a N α subset of R . Set α0 = sup{α > 0 : H (S) = ∞}. Also com- α pute α1 = inf{α > 0 : H (S) = 0}. Then α0 ≤ α1. This number is called the Hausdorff dimension of S. What is the Hausdorff dimension of the Cantor ternary set? What is the Hausdorff dimension of a regularly imbedded, k-dimensional, C1 manifold in RN ? In fact (see [FOL, p. 325], [FE1]), any rectifiable set N S ⊆ R has the property that α0 = α1.

The measure HN−1 gives one reasonable definition of dσ on ∂ when ⊆ RN has C1 boundary. Now let us give another. If S ⊆ RN is closed and x ∈ RN , let dist (x, S) = inf{|x − s| : s ∈ S}. Then dist (x, S) is finite, and there is a (not necessarily unique) s0 ∈ S with |s0 − s| = dist (x, S). (Exercise: Prove these assertions.) Suppose that M ⊆ RN is a regularly imbedded C1 manifold of dimension k < N (see [HIR]). Let E ⊆ M be compact and, for > 0, set E = {x ∈ RN : dist (x, E) < }. Define

LN (E ) = σk(E) lim sup N−k , →0+ MN−k where LN is Lebesgue volume measure on RN . It can, in fact, be shown that “limsup” may be replaced by “lim.” The resulting set-function σk k is an outer measure. It can be proved that H (E) = σk(E). The measure σk may be extended to more general subsets of M by the usual exhaustion procedures. Our third deﬁnition of area measure is as follows. Let M ⊆ RN be a regularly imbedded C1 submanifold of dimension k < N. Let p ∈ M, and let (ψ, U) be a coordinate chart for M ⊆ RN , as in the deﬁnition of “regularly imbedded submanifold.” We use the notation

JRG to denote the Jacobian matrix of the mapping G. 322 Appendices and Ancillary Material

When E ⊆ U ∩ M is compact, deﬁne

Z = h −1 −1 i k mk(E) vol JRψ (x)e1,..., JRψ (x)ek dL (x). ψ(E)

th Here e j is the j unit coordinate vector, and the integrand is simply the k-dimensional volume of the k-parallelipiped determined by the −1 = vectors JRψ (x)e j , j 1,..., k. We know from calculus [SPI] that this gives a definition of surface area on compact sets E ⊆ U ∩ M that coincides with the preceding definitions. The new definition may be extended to all of M with a partition of unity, and to more general sets E by inner regularity. Finally, we mention that a k-dimensional, C1 manifold M may be given (locally) in parametrized form. That is, for P ∈ M there is a N neighborhood P ∈ UP ⊆ R and we are given functions φ1, . . . , φN k defined on an open set WP ⊆ R such that the mapping

8 = (φ1, . . . , φN ) : WP → UP ∩ M is C1, one-to-one, and onto, and the Jacobian of this mapping has rank k at each point of WP . In this circumstance, we define Z k τk(UP ∩ M) = |Mx | dL (x), x∈WP where Mx is defined to be the standard k-dimensional volume of the k image of the unit cube in R under the linear mapping JR8(x). [The object Mx can be defined rather naturally using the language of differential forms—see [SPI]. The definition we have given has some intuitive appeal.]

On a k-dimensional regularly imbedded C1 submanifold of RN , we have

k H = σk = mk = τk. Appendices and Ancillary Material 323

Appendix V Green’s Theorem

Here we record the standard form of Green’s theorem that is used in harmonic analysis. A derivation of this particular formula from Stokes’s theorem appears in [KRA4, Section 1.3]; that reference also contains applications to the theory of harmonic functions. See also [BAK], [KRP].

Theorem: Let ⊆ RN be a domain with C2 boundary. Let dσ denote area ((N − 1)-dimensional Hausdorff) measure on ∂—see Appendix IV. Let ν be the unit outward normal vector ﬁeld on ∂. Then, for any functions u, v ∈ C2(),¯ we have

Z ∂v ∂v Z u − v dσ = (u1v − v1u) dV. ∂ ∂ν ∂ν

Appendix VI The Banach-Alaoglu Theorem

Let X be a Banach space and X ∗ its dual. Assume for the moment that ∗ X is separable. For φ j , φ elements of X , we say that φ j → φ in the weak-∗ topology if, for each x ∈ X, φ j (x) → φ(x). Thus weak-∗ convergence is pointwise convergence for linear functionals. It induces the weakest topology on X ∗ under which the point evaluation functionals are continuous.

Theorem: Assume that the Banach space X is separable. Let {φ j } be ∗ { } a bounded sequence in X . Then there is a subsequence φ jk that con- verges in the weak-∗ topology.

Informally, we often cite the Banach-Alaoglu theorem by saying that “the unit ball in the dual of a Banach space is weak-∗ compact.” 324 Appendices and Ancillary Material

Note that the unit sphere in the dual of a (inﬁnite-dimensional) Banach space is never weak-∗ compact (e.g., take an orthonormal sequence in Hilbert space). In case the Banach space X is not separable, then we cannot specify the weak-∗ topology using sequences; we must instead specify a sub-basis for the topology. See [RUD3] for the details.

Appendix VII Expressing an Integral in Terms of the Distribution Function

Let f be a measurable function on RN . For α > 0 we set N µ f (α) = m{x ∈ R : | f (x)| > α}.

Then µ f is the distribution function of f . Integrals of the form R φ(| f |) dV may be expressed in terms of integrals of µ (α). For RN f example, Z Z ∞ p p−1 | f (x)| dx = µ f (α)pα dα. RN 0 For a proof, test the assertion on f the characteristic function of an interval, and then use standard approximation arguments.

Appendix VIII The Stone-Weierstrass Theorem

Let X be a compact metric space. Let C(X) be the algebra of continuous functions on X equipped with the supremum norm. Let A ⊆ C(X) be an algebra of continuous functions that contains the constant function 1. [Here an algebra is a vector space with a notion of multiplication—see [LAN].] The Stone-Weierstrass theorem gives conditions under which A is dense in C(X). Appendices and Ancillary Material 325

Theorem: Assume that, for every x, y ∈ X with x 6= y, there is an f ∈ A such that f (x) 6= f (y) (we say that A separates points). [In case the functions in A are complex-valued, f ∈ A whenever f ∈ A.] Then A is dense in C(X).

Appendix IX Landau’s O and o Notation

Sometimes a good piece of notation is as important as a theorem. Lan- dau’s notation illustrates this point. Let f be a function deﬁned on a neighborhood in RN of a point P. We say that f is O(1) near P if

| f (x)| ≤ C, for some constant C, in that neighborhood of P. Now let g be another function deﬁned on the same neighborhood of P. Writing this last inequality somewhat pedantically as

| f (x)| ≤ C · 1, we are led to deﬁne f to be O(g) near P if

f (x) ≤ C · g(x) , for some constant C, in that neighborhood of P. If instead, for any > 0, there is a δ > 0 such that

f (x) ≤ · g(x) when 0 < |x − P| < δ, then we say that f = o(g) at P. We will see ample illustration of the Landau notation in our study of Fourier analysis.

Table of Notation

Section Notation Meaning 0.2 R real numbers 0.2 (a, b), [a, b)(a, b], [a, b] intervals

0.2 χS characteristic function 0.2 s(x) simple function 0.2 RN Euclidean space 0.2 R f (x) dx Lebesgue integral 0.2 f + positive part of function 0.2 f − negative part of function 0.2 LMCT Lebesgue monotone convergence theorem 0.2 LDCT Lebesgue dominated convergence theorem 0.2 FL Fatou’s lemma y 0.2 fx , f slice functions 0.2 L p Lebesgue space 0.2, 0.3 T circle group 0.2 A a σ-algebra

327 328 Table of Notation

Section Notation Meaning

0.2 δ0 Dirac measure 0.2 µ a measure 0.3 X a Banach space 0.3 k k a norm p 0.3 k kp , k kL p L norm

0.3 k k∞ , k kL∞ essential sup norm p p 0.3 Lloc local L space 0.3 k kX Banach space norm 0.3 L : X → Y linear mapping of spaces

0.3 k kop operator norm 0.3 X ∗ dual space of X 0.3 L∗ adjoint of L 0.4 h · , · i inner product 0.4 k k Hilbert space norm 0.4 H Hilbert space 0.4 ⊥ perpendicular or orthogonal 0.4 7→ maps to 0.4 K ⊥ set of vectors perpendicular to (annihilator of) K 0.4 A + B linear sum of spaces 0.4 `2 little Lebesgue space 0.5 FAPI Functional Analysis Principle I 0.5 FAPII Functional Analysis Principle II

1.1 ∂t u partial derivative in t

1.1 ∂x u partial derivative in x Table of Notation 329

Section Notation Meaning 1.1 f (k) kth derivative 1.1 S unit circle in the complex plane

1.1 τg translation by g ∗ 1.1 C the multiplicative group C \{0} 1.1 bf ( j) jth Fourier coefﬁcient ∞ P i jt 1.1 S f ∼ bf ( j)e formal Fourier expansion j=−∞ k ∞ 1.1, App I Cc , Cc compactly supported, smooth functions

1.2 SN f partial sum of Fourier series

1.2 DN (s) Dirichlet kernel 1.2 f ∗ g convolution of functions 1.2, App IX O(t) Landau’s notation 1.3 f (x+) right limit of f at x 1.3 f (x−) left limit of f at x 1.3 3 sequence of scalars

1.3 M3 multiplier operator induced by 3

1.3 σN f Cesaro` mean of f

1.3 K N Fejer´ kernel

1.3 Pr Poisson kernel for the disc

1.3 Pr f Abel sum of f

1.4 {kN } a family of kernels 1.4 A ≈ BA is comparable in size to B 1.5 M f Hardy-Littlewood maximal function 330 Table of Notation

Section Notation Meaning 1.5 P∗ f maximal Poisson integral of f 1.6 d( f, g) metric induced by L2 topology 1.6, App VIII C(T) set of continuous functions on T 1.6 sgn x signum function 1.6 P.V. R dx Cauchy principal value 1.6 H f Hilbert transform of f 1.6 He f modiﬁed Hilbert transform of f 1.6 J f complexiﬁed Hilbert transform of f th 1.6 e j the j exponential function 1.6 f ∗ K convolution of a function with a kernel 2.1 t · ξ inner product on Euclidean space 2.1 bf (ξ) Fourier transform 2.1 ()b Fourier transform 2.1 much greater than

2.1 C0 continuous functions that vanish at ∞ 2.2 O(N) orthogonal group on RN 2.2 ρ a rotation δ 2.2 αδ , α dilation operators

2.2 τa translation operator 2.2 ef odd reﬂection of f Table of Notation 331

Section Notation Meaning

2.3 G Gaussian summability kernels 2.3 gˇ(x) inverse Fourier transform 2.3 F Fourier transform 2.4 z complex variable 2.4 Var f variance of f 2.4 [A, B] commutator of A and B

3.1 eSN f alternative method of partial summation 3.1 T2 torus group 3.1 |( j, k)| modulus of an index sph 3.1 SR f spherical partial sum sq 3.1 SM square partial sum rect 3.1 S(m,n) rectangular partial sum poly,P 3.1 SR polygonal partial sum 3.3 B(0, 1) unit ball in Euclidean space

3.3 MB multiplier operator for the ball

3.3 Mm multiplier operator induced by m N 3.3 Q N fundamental region in T 2 3.3 η(x)(2π)−N/2e−|x| /2 √ −N/2 N/2 −|x|2/2 3.3 η (x)(2π) e 3.3 3 fundamental lattice in RN 3.3 eS f periodized multiplier operator induced by S 3.4 Q unit cube in RN 332 Table of Notation

Section Notation Meaning

3.4 Q R dilate of unit cube 3.4 Hφ Hilbert transform of φ

3.4 Ev half space determined by vector v 3.5 4ABC triangle with vertices A, B, C 0 3.5 TA triangular sprouts 3.5 Tjk` sprouted triangles k 3.5 6k unit sphere in R 3.5 T j half space multiplier operator 3.5 Re adjunct rectangles to R 3.5 Re subadjunct rectangles to R 3.5 |E| Lebesgue measure of the set E iθ 4.1 fr (e ) circular slice of f th 4.1 Fj,r j Fourier coefﬁcient of fr

4.1 Hk spherical harmonics of degree k

4.1 ρφ rotation through angle φ

4.1 α = (α1, . . . , αN ) multi-index 4.1 |α| modulus of a multi-index 4.1 xα multi-index product notation ∂α 4.1 ∂xα multi-index derivative notation 4.1 α! multi-index factorial notation Table of Notation 333

Section Notation Meaning

4.1 δαβ Kronecker delta

4.1 Pk space of homogeneous polynomials of degree k

4.1 dk dimension of Pk 4.1 P(D) differential polynomial 4.1 hP, Qi inner product on homogeneous polynomials 4.1 4 Laplace operator

4.1 ker φk kernel of φk

4.1 im φk image of φk

4.1 Ak space of solid spherical harmonics of degree k 4.1 Y (k) a spherical harmonic of degree k 4.1 ∂/∂ν unit outward normal derivative 4.1 σ, dσ rotationally invariant surface measure

4.1 ak dimension of the space Hk

4.2 ex0 point evaluation functional (k) 4.2 Zx0 zonal harmonic 4.2 P(x, t0) Poisson kernel for the ball λ 4.2 Pk (t) Gegenbauer polynomial th 4.2 Jk k Bessel function 5.1 D2 j (2 j)th-order differentiation operator 2 j 5.1 k2 j kernel associated to D 334 Table of Notation

Section Notation Meaning 5.1 I2 j (2 j)th-order integration operator 5.1 Iβ βth-order fractional integration operator 5.2 K (x) Calderon-Zygmund´ kernel 5.2 (x)/|x|N Calderon-Zygmund´ kernel

5.2 TK Calderon-Zygmund´ singular integral operator 5.2 (x) numerator of Calderon-´ Zygmund kernel 5.2, 5.8 BMO functions of bounded mean oscillation 5.4 D unit disc in the complex plane 5.4 H p(D) pth-order Hardy space on D 5.4 H ∞(D) space of bounded holomorphic functions 5.4 f ∗ boundary limit function of f 5.4 hp(D) pth-power integrable harmonic functions on D 5.4 h∞(D) bounded harmonic functions on D

5.4 Ba(z) Blaschke factor 5.4 B(z) Blaschke product

5.4 BK partial Blaschke product 5.4 B∗ boundary function of a Blaschke product B Table of Notation 335

Section Notation Meaning 5.5 v conjugate harmonic function to the harmonic function u 5.5 φe boundary function for v 5.5 ek(t) = cot(t/2) Hilbert transform kernel on disc 1 5.5 HRe real-variable Hardy space of order 1 th 5.5 K j (x) j Riesz kernel th 5.5 R j j Riesz transform

5.5 k k 1 real-variable Hardy space HRe norm

5.5 Py(x) Poisson kernel for the upper half-space 5.6 M f Hardy-Littlewood maximal operator 5.6 f ∗ maximal operator associated to the kernel φ0 5.7 a(t) atom p 5.7 HRe real-variable Hardy space of order p

5.7 k k∗ BMO norm 5.7 Q cube in RN 6.1 ρ(x, y) quasi-metric 6.1 B(x, r) ball determined by a quasi-metric 6.1 (X, ρ, µ) space of homogeneous type

6.1 C1, C2 constants for space of homogeneous type 336 Table of Notation

Section Notation Meaning 6.1 ρ(z, w) nonisotropic distance on the unit ball of Cn 6.1 Mx presentation of the Heisenberg group

6.2 Rα(x) kernel of fractional integral operator x 6.2 Ey , E partial distribution functions 6.2 M f (x) Hardy-Littlewood maximal function 6.2 M∗ f (x) modiﬁed Hardy-Littlewood maximal function S 6.2 O = j B(o j , r j ) Whitney decomposition 6.2 δO distance to the complement of O P 6.2 f = g + j h j Calderon-Zygmund´ decomposition

6.2 Tk integral operator induced by k

6.2 K N truncated kernel 6.3 a(x) atom 1 6.3 HRe real-variable or atomic Hardy space 6.3 T (1) theorem on L2-boundedness of integral operators 6.3 T t transpose of the operator T 6.3 T (b) theorem generalization of the T (1) theorem Table of Notation 337

Section Notation Meaning 6.3 λ( j) coefﬁcients in the Cotlar- Knapp-Stein theorem 6.3 φx, translate and dilate of φ 7.3 MRA Multi-Resolution Analysis 7.3 φ wavelet scaling function 7.3 ψ wavelet

7.3 Vj subspaces in an MRA decomposition

7.3 W j wavelet subspaces

7.3 MRA1 scaling axiom for an MRA

7.3 MRA2 inclusion axiom for an MRA

7.3 MRA3 density axiom for an MRA

7.3 MRA4 maximality axiom for an MRA

7.3 MRA5 basis axiom for an MRA

7.4 fN approximation to the Dirac delta

7.4 DeN (t) sin Nt/πt 7.5 m(ξ) low-pass ﬁlter

App II ρα,β Schwartz semi-norms App II S the Schwartz space k App IV Hδ approximate Hausdorff measure App IV Hk Hausdorff measure

App IV α0, α1 lower and upper Hausdorff dimensions

App IV E -thickening of the set E 338 Table of Notation

Section Notation Meaning

App IV σk k-dimensional measure deﬁned by thickening

App IV mk k-dimensional measure deﬁned by pullback

App IV τk k-dimensional measure by parametrization App VI weak-∗ topology pointwise topology on a dual space

App VII µ f distribution function of f App VIII C(X) algebra of continuous functions on X App VIII A subalgebra of C(X) App IX O, o Landau’s notation Bibliography

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Index

Abel summation, 58 space, 14 absolutely continuous measure, 13 spaces, examples of, 15 abstract measure theory, 12 -Steinhaus theorem, 17 adjoint of a bounded linear operator, subspace, 16 20 basis axiom for an MRA, 285 almost Bernoulli diagonalizable operator, 309 dream, 47 everywhere, 5 solution of wave equation, 33 annihilator of S, 23 Besicovitch, A., 148 atomic Bessel decomposition of a Hardy space functions, 197 function, 229 inequality, 25, 79 theory of Hardy spaces, 227 Blaschke atoms factor, 214 definition of, 227 factorization, 214, 217 and functions of bounded mean product, 214 oscillation, pairing of, 233 product, boundary limit function of, on a space of homogeneous type, 218 263 product, convergence of, 216 on a space of homogeneous type BMO functions for small p, 266 and L p, 233 of type (1, ∞), 263 as a substitute for L∞, 234 axioms boundary of Mallat for an MRA, 285 behavior of harmonic functions, for a Multi-Resolution Analysis, 213 285 limits of Hardy space functions, 210 Baire category theorem, 66 bounded Banach domain, boundary of, 237 -Alaoglu theorem, 211, 323 linear functional, 17 347 348 Index bounded mean oscillation characters equivalent definitions of, 230 of the circle group, 43, 44 function of, 208, 230 of the circle group as a complete norm for, 231 orthonormal system, 78 boundedness of a group, 96 of the Hardy-Littlewood maximal of a locally compact abelian group, operator on L p, 73 44 N of the square multiplier, of R , 96 independent of scale, 141 Chebyshev’s inequality, 11 Brelot potential theory, 235 circle group (T), 12, 43 and the unit circle in the plane, 43 calculus of operators, 208 closure Calderon,´ A., xiv, 144, 203, 208, 231 axiom for an MRA, 285 commutators, 272 of measurable functions under sup, Calderon-Zygmund´ inf, lim sup, lim inf, 5 decomposition, 255 of measurable sets under kernels, 208 complementation, 3 kernels, truncated, 257 of measurable sets under countable lemma, 253 intersection, 3 operators acting on BMO, 233 of measurable sets under countable operators acting on H1, 264 union, 3 operators and the T (1) theorem, coefficient region, 125 272 commutation with translations, 43 theorem, 207 complete orthonormal system in L2, theorem for a space of 181 homogeneous type, 256, 257 complex canonical factorization, 217 normal direction, 240 Cantor middle thirds set, 3 tangential direction, 240 Caratheodory’s´ construction, 320 unit ball, boundary of as a space of Carleson, L., 92 homogeneous type, 239 Carleson-Hunt theorem, 56, 71, 92 conjugate Cauchy exponent, 9 integral formula, 210 index, 8 kernel, imaginary part of, 220 Poisson kernel, 221 principal value, 86, 206 system of harmonic functions, 224 -Riemann equations, generalized, construction of a wavelet, 306 223, 224 continuous wavelets, construction of, -Schwarz-Bunjakowski inequality, 303 9 convergence -Schwarz inequality, 9, 22 failure of for continuous functions, Cesaro´ summation, 57 66 character group of Fourier series in L p norm, 77 N 2 of R , 95 in L of Fourier series, 78 of T, 46 convolution characteristic function, 5 and Fourier transform, 103 Index 349

N of functions on R , 12 directionally limited metric space, on T, 12 251 Cordoba, A., 170 Dirichlet cosine wave, approximation of, 290 and convergence of series, 34 Cotlar, M., 267 kernel, 51 Cotlar-Knapp-Stein theorem, 258, L1 norm of, 62, 77 267, 269 problem on the ball, 187 countable distribution additivity, 3 function, 324 subadditivity, 3 theory, 267 counting argument, 174 doubling property, 236, 237 covering lemma for a noncompact set, dual 248 of a Banach space, 18 critical indices for homogeneous of H1 and BMO, 231, 232, 233 functionals, 205 of L∞, 19 N cubes in R , 230 of the Lebesgue spaces, 19 d’Alembert solution of vibrating electrocardiogram software, 273 string, 32 equivalence classes of measurable Daubechies, I. functions, 5 construction of a wavelet basis, 297 Euclidean space wavelets and Fourier analysis, 304 as a space of homogeneous type, wavelets, combinatorial 237 construction of, 302 nonisotropic, 237 David, G., 266, 271 exponential map, 238 David-Journe´ theorem, 271 decay of Fourier coefﬁcients and failure smoothness, 49 of boundary limits in L1, 212 decomposition of the Dirichlet kernels to form a of a function on all of space, 171 standard family, 77 2 of L into V j ’s and W j ’s, 283 Fatou’s lemma (FL), 7 dense subspace, 20 FAWAV, 293 density axiom for an MRA, 285 Fefferman’s counterexample derivative ﬁnal calculation in, 166 Fourier transform of, 96 logic of, 170 and the Fourier transform, 199 Fefferman, C., 71, 118, 148 differentiation, fractional, 199 Fefferman, R., 170 dilations, 99 Fefferman-Stein Fourier transform and, 99 decomposition, 232 dimensional analysis, 202 theorem, 264 Dirac delta mass Fejer´ Fourier series expansion of, 278 kernels as a standard family, 63 Fourier transform of, 277 summation, 60 spectral analysis of, 277 summation, kernel of, 57 direct sum decomposition of L2, 173 summation method, formula for, 57 350 Index

Fejer´ (continued) series, convergence of, 50 summation method, multiplier for, series, mathematical theory, 34 58 series, norm convergence from theorem, 55 modern point of view, 87 fiber-optic communication, 276 series, partial sum of, 50 finite volume of balls, 237 series, summability of, 47 Folland, G. B., 196, 276 solution of the heat equation, 35, 37 Folland-Stein fractional integration transform, 95, 317 theorem, 242 transform, action on L p, 116 Fourier analysis transform and conjugation, 101 and complex variables, 171 transform and convolution, 103 custom, 274 transform and homogeneity, 103 designer, 275 transform and odd reflection, 101 of the Dirac mass, 286 transform of characteristic function in Euclidean space, 95 of rectangle, 162 on locally compact abelian groups, transform, derivative of, 97 44 transform, eigenfunctions of, 116 a new look, 171 transform, eigenvalues of, 115 Fourier, Joseph, 34 transform, invariance under group coefficients, 46 actions, 95 coefficients of a smooth function, transform, invariant spaces of, 173, size of, 48 197 deriviation of the formula for transform, inverse of, 104 coefficients, 38 transform on L1, 96 expansion, formal, 47 transform on L2, 115 integral operators, 209 transform, mapping properties of, integrals, quadratic theory, 79 116 inversion formula, 108, 110, 111 transform, nonsurjectivity of, 113 multiplier, 56 transform, quadratic theory, 114 multiplier for L1, 132 transform, sup norm estimate, 96 multiplier for L2, 131 transform, surjectivity of, 112 multiplier for L p, 131 transform, uniform continuity of, multiplier, periodization of, 134 98 multipliers and convolution kernels, transform, univalence of, 112 84 transform, weak, 103 multipliers and summation of Treatise on the Theory of Heat, 35 Fourier series, 133 fractional integral multipliers and summation of integrals, mapping properties of, multiple Fourier series, 137, 203 139, 140 operator, 201, 242 multipliers as L∞ functions, 133 fractional integration, 201 multipliers, self-duality of, 132 on a space of homogeneous type, series, 39 245 series, convergence at a point of theorem of Riesz, 202 differentiability, 52 frequency modulation, 276 Index 351

Frobenius theorem, 238 wavelet subspaces, 284 Fubini’s theorem, 8 Hahn-Banach theorem, 18 function Han, Y. S., 263 spaces, 14 Hardy, G. H., 209 what is, 33 Hardy spaces, 221 Functional Analysis atoms for small p, 228 Principle I, 27 on the disc, 210 Principle I and norm convergence, function, boundary limit of, 219 28 H p for p < 1, 224 Principle I, proof of, 28 Hardy-Littlewood maximal Principle II, 27 function, 213, 225, 246 Principle II and pointwise maximal function deﬁnition, 226 convergence, 28 measurability of, 71 Principle II, proof of, 29 and the Riesz transforms, 263 functions real variable theory, 231 of bounded mean oscillation, 229 unboundedness on L1, 74 calculus-style, 55 weak-type estimate for, 72 with values in the extended reals, 5 harmonic analysis, axiomatic theory for, 235 gamma function, 197 harmonic conjugate, 219 Gauss-Weierstrass boundary limit of, 220 kernel, 106 Hausdorff kernel, Fourier transform of, 107 measure, 320 summation, 108 space, 241 Gaussian heat equation, 36 Fourier transform of, 104, 105 derivation of, 35 integral, 105 heated rod, 35 kernel, 104 Heisenberg Gegenbauer polynomial, 194 group, 240 geometric sets as models for uncertainty principle, 117 summation regions, 133 Hermite polynomials, 116 gradient of a harmonic function, Hermitian inner product characterization of, 223 on homogeneous polynomials, 175 Green’s theorem, 179, 323 positive-deﬁnite, 21 groups that act on Euclidean space, 99 Hilbert space, 22 element, Fourier representation of, Haar 26 basis, shortcomings of, 300 examples of, 23 expansion vs. Fourier expansion of isomorphism with `2, 26 a cosine wave, 293 operators, summing, 268 series expansion of the Dirac delta separable, 25 mass, 286 summability, 26 series expansion of a truncated Hilbert subspace, 23 cosine wave, 291 Hilbert transform, 81, 203, 207, 221 wavelet basis, 279 boundedness of, 28 352 Index

Hilbert transform (continued) injectivity of L and dense range of boundedness using the Cotlar- L∗, 20 Knapp-Stein theorem, 269 inner product space, 21 complexified version, 91 examples of, 21 interpretation in terms of analytic real, 95 functions, 91, 92 inner regularity, 4, 73 interpretation in terms of harmonic integrable functions, 7, 15 conjugates, 92 integral kernel of, 86 of a complex-valued function, 6 on L2, 82 in terms of distribution function, on L p, 82 324 matrix of, 311 Lebesgue, 7 modified, 88, 89 of a measurable function, 6 as a multiplier operator, 81 of a non-negative measurable and norm convergence of Fourier function, 6 series, 87 Riemann, 7 and partial sums, 81 of a simple function, 6 significance of, 81 interpolation, as a singular integral, 203 complex method of, 318 unboundedness on L1 and L∞, 89 real method of, 318 homogeneity interval, 2 and the Calderon-Zygmund´ inverse Fourier transform, 104, 111, theorem, 256 131 of functions and distributions, 102 iterated integrals, 8 of a singular integral kernel, 203 homogeneous Jensen’s of degree β, 102 formula, 215 distributions, 205 inequality, 9, 215 polynomial, decomposition of, John, F., 229 176 John-Nirenberg inequality, 231, polynomials, space of, 174 233 homomorphisms from the circle Journe,´ J.-L., 266, 267, 272 group into the unit circle, 44 Hunt, R., 92 Kakeya, S., 148 theorem, 92 needle problem, Besicovitch hyperbolic distance, 311 solution of, 151 Holder’s¨ inequality, 9, 16 Kohn, J. J., 209 Hormander¨ Kohn/Nirenberg calculus, 209 hypoellipticity condition, 238 Kolmogorov, A., 92 example, 93 image compression, 277 inclusion axiom for an MRA, 285 L1 is not a dual, 19 indicator function, 5 L2 as a Hilbert space, 23 infinite-dimensional spaces, 14 L p spaces, 8, 15 infinite products, 216 Lagrange interpolation, 34 Index 353

Landau’s notation, 325 mean-value zero condition, 203 Laplacian, 223 and Hormander¨ condition, 260 Lebesgue for atoms, justification, 228 differentiation theorem, 11, 249, higher order, 228 250 measurability, 3 dominated convergence theorem measurable (LDCT), 7 function, 5 N integration on R , 13 set, 2 measure, 2 measure monotone theorem (LMCT), 7 of an interval, 3 points, 11 of a set, 2 spaces, 15 with weight φ, 13 spaces, infinite dimensionality of, medicine, 276 16 Meyer’s lemma, 160 spaces, local, 15 Minkowski, H. Lebesgue-Tonelli measure, 2 inequality, 9, 15 length, 2 integral inequality, 10, 16 Lewy unsolvable operator, 241 modified linear Hardy-Littlewood operator, functional, 17 weak-type estimate for, 249 functional, null space of, 25 Hilbert transform, 88 operator, bounded, 16 molecules, 229 operator, continuity of, 17 moment conditions for wavelets, localization 301 in the space variable, 276 Morera’s theorem, 204 in the time/phase variable, 276 Moser, J., 230 low-pass filter, 304 MRA, 279 Lusin conjecture, 92 multi-index notation, 174 Multi-Resolution Analysis, 276 Macias, R., 263 multivariable versus one-variable Mallat, S., 279 Fourier analysis, 122 Marcinkiewicz, S multiplier interpolation theorem, 73, 318 associated with the Fejer´ multiplier theorem, 59 summation method, 58 martingale theory, 230 operator for the ball, 131 maximal function operators on L2, 80 associated to a family of operators, for partial summation, 80 27 for Poisson summation, 58 modeled on a smooth test function, problem for ball, counterexample 226 to, 168 maximality axiom for an MRA, problem for the half-space, 144 285 problem for the square and maximal Poisson operator, summation of multiple Fourier majorization by Hardy- series, 143 Littlewood operator, 75 problem for the square/cube, 143 354 Index multiplier (continued) matrix, 99 theorem for the cube and projection on a subspace, 25 summation of multiple Fourier orthogonality, 23 series, 140 of the sine functions, 38 music recording, 276 of trigonometric basis functions, 38 orthonormal basis, 25 nearest element in a subspace, 25 complete, 25 Nirenberg, L., 209, 229 countable, 25 non-existence of C∞, compactly for L2, 78 supported wavelets, 297 oscilloscope analysis, 279 nonisotropic outer balls, 239 measure, 2 dilations, 241 regularity, 4 norm outward normal, unit, 179 associated to an inner product, 22 convergence and FAPI, 69 pth-power integrable functions, 7 convergence and Fejer´ means, 70 para-accretive functions, 272 convergence and standard families parallelogram law, 24 of kernels, 69 parallels orthogonal to a vector, 190 convergence, application of duality paraproducts, 272 to, 90 Parseval’s formula, 79 convergence, failure in L1 and L∞, partial differential equations, 272 89 partial sum of Fourier series, closed convergence of Fourier series, 69 formula for, 51 convergence in L2, 81 partial summation, 60 convergence in L p, 84 alternative method, 122 convergence of partial sums of and Fourier multipliers, 130 Fourier series, 76 numerical examples of, 126 limits of Poisson integrals, 212 in several variables, 122 on a finite-dimensional vector partial sums space, uniqueness of, 14 alternative definitions for, 121 on a vector space, 13, 22 and the Hilbert transform, 121 summability, failure of in L1 and Peter-Weyl theorem, 79 L∞, 91 physical principles governing heat, 35 summation of multiple Fourier Plancherel formula, 114 series, 140 polarized form, 115 normed linear space, Poincare´ metric, 311 finite-dimensional, 13 point evaluation functional, 183 point mass at the origin, 13 open mapping principle, 17 pointwise convergence operator norm, 17 of Borel measures, 212 optics, 276 failure in L1, 93 Orlicz spaces, 266 of Fejer´ summation, 76 orthogonal of Fourier series at a point of group, 189 differentiability, 52 Index 355

of the Poisson sums of Fourier real-variable Hardy space, 222, 223 series, 76 recovering a continuous function for the standard summability using a summability method, 66 methods, 71 rectangular convergence Poisson integral formula, 210 restricted, 123 and spherical harmonics, 182 unrestricted, 124 Poisson kernel, 59, 221 rectangular summability, unrestricted, for the ball, 186 124 on the ball and spherical reflexive space, 19 harmonics, 187 regular Borel measure, 236 majorization by the regularity of Lebesgue measure, 4 Hardy-Littlewood operator, 74 relationship of V j to W j , 284 as a standard family, 63 representation theory, 174 in terms of spherical harmonics, reproducing property of the zonal 181, 187 harmonics, 184 Poisson restricted maximal operator, weak-type rectangular convergence, 147 estimate for, 75 rectangularly convergent but summation, 58, 60 spherically divergent, 129 summation, kernel of, 59 rectangularly convergent but summation of several variables, unrestrictedly rectangularly 133, 134 divergent, 129 polarized Plancherel formula, 102 rectangular summability, 124 polygonal summation for multiple restriction to unit sphere of harmonic Fourier series, 147 polynomials, 173 positivity of the Fejer´ and Poisson Riemann-Lebesgue lemma, 48, 97 kernels, 63 intuitive view, 98 probability theory, 230 Riesz product measurable functions, 8 fractional integration theorem, 245 projection in a Hilbert space, 24 fractional integration theorem, properties of measurable sets, 4 modern interpretation, 245 pseudodifferential operators, calculus operators, 242 of, 208 representation theorem, 24 theorem on L p boundedness of the quadratic theory of Fourier series, 79 Hilbert transform, 87 quantum mechanics, 240 theory of homogeneous functionals, quasi-metric, 236 204 quasi-triangle inequality, 236, 246 transforms, 222 transforms, canonical nature of, radial 222 boundary limits on the disc, 214 transforms, kernels of, 222 function, Fourier transform of, 197, Riesz-Fischer theorem, 79 198 Riesz-Thorin interpolation theorem, radio recording, 276 116, 318 real line, 2 rotation group, action of, 171 356 Index rotationally invariant surface measure, smooth wavelets, construction of, 306 178 Sobolev rotations, 99 imbedding theorem, 224 and the Fourier transform, 99 spaces, 266 solid spherical harmonics, 177 σ-algebra, 3, 12 action of the Fourier transform on, scaling axiom for an MRA, 285 198 scaling function φ, 279 space of homogeneous type, 229, 236, Schauder basis, 308 239 Schur’s lemma, 67, 244 examples, 237 and functional analysis principle I, spaces, nonlocally convex, 16 68 special orthogonal matrix, 99 Schwartz spectral analysis, 276 functions, 317 spherical harmonics, 174, 177, 178 kernel theorem, 84, 200 density in L2, 178 Segovia, C., 263 expansion, 177 self-duality of Fourier multipliers, 132 solid, 178 semicontinuous, lower, 246 surface, 178 semi-norms on the Schwartz space, spherical summation, 122, 147 317 spherically convergent but square Semmes, S., 267, 272 divergent, 127 separable, 25 spikes in audio recordings, 279 set-function, 12 sprouting of triangles, 152 sets of measure zero, 3, 4 square convergent several complex variables, 272 but restrictedly rectangularly signal divergent, 128 compression, 277 but spherically divergent, 128 processing, 277 square summation simple function, 5 of multiple Fourier series, 146 sines and cosines, inadequacy of, 274 pathologies of, 147 singular integral standard acting on atoms, 264 families and convergence of Fourier boundedness on L2, 267 series, 65 kernel, 203 family of summability kernels, 63 L p boundedness of, 207 kernel, 270 on Lipschitz curves, 272 Stein, E. M., 231 on a space of homogeneous type, Stein-Weiss definition of Hardy 257 spaces, 223 on spaces of homogeneous type, Stieltjes integral, 10 L p estimates, 260 Stone-Weierstrass theorem, 78, 317, operator, 203 324 size of Fourier coefficients, norm strong-type estimate, 28 estimate of, 47 strongly singular integrals, 205 Sjolin,¨ P., 92 subharmonicity and Hardy spaces, theorem, 92 210 Index 357

subspaces V j in an MRA unconditional basis, 297, 307, 308 decomposition, 281 constructed with wavelets, 307 summability kernels, uniform boundedness principle, 17 majorization by the Hardy- unit ball in complex space, 239 Littlewood maximal operator, unit disc, 210 74 unrestrictedly rectangularly properties of, 62 convergent but spherically summability methods divergent, 129 for Fourier series, 56 summary of, 59 vanishing moments and oscillation, summation of Fourier series at a 301 Lebesgue point, 139 variance, 119 support of a function, 315 vector space, 13 surface measure, 178 vibrating string, 31 via parametrization, 322 via pullback, 321 Walker, J., 105, 276, 289, 293 via thickening, 321 wave equation, 32 surjectivity, failure of for Fourier derivation of, 39 transform, 98 wavelets, 114 symmetries of Euclidean space, and almost diagonalizable 235 operators, 309 basis, 284 T (1) theorem, 271, 272 as ﬂexible units of harmonic T (b) theorem, 272 analysis, 275 Taibleson, M., 92 function ψ, 280 telecommunications, 276 weak-type television recording, 276 bound for an operator, 72 testing functions, 315 function, 318 density in L p, 316 inequality, 28 Toeplitz, O., 209 weak-∗ convergence, 324 torus group, 122 weakly bounded operator, 270 translation-invariant operators, 101, weight, 13 200, 275 Weiss, G., 148 and kernels, 101 Whitney decomposition of an open translations, 99, 100 set, 252 Fourier transform and, 100 Wiener covering lemma, 71, 246 triangle inequality for an inner product space, 22 zeros of a Hardy space function, 215, trigonometric polynomials, 46 216 zonal harmonics, 183, 195 Uchiyama, A., 263 explicit formula for, 189 uncertainty principle, 117 in terms of Gegenbauer qualitative version, 117 polynomials, 195, 196 quantitative version, 118 Zygmund, A., xiv, 144, 203, 208, 231 and wavelets, 120 lemma, 159 AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS

A PANORAMA OF HARMONIC ANALYSIS

STEVEN G. KRANTZ

Steve Krantz has written a remarkable book that leads the reader on a tour of Euclidean harmonic analysis from its genesis to the frontiers of recent research. It can be warmly recommended to professionals, graduate students, and adventurous undergraduates who want to know what this beautiful subject is about. —Gerald Folland, University of Washington, Seattle

This book treats the subject of harmonic analysis, from its earliest beginnings to the latest research. Following both a historical and a conceptual genesis, the book discusses Fourier series of one and several variables, the Fourier transform, spherical harmonics, fractional integrals, and singular integrals on Euclidean space.

The climax of the book is a consideration of the earlier ideas from the point of view of spaces of homogeneous type. The book culminates with a discussion of wavelets—one of the newest ideas in the subject.

The book is intended for graduate students, advanced undergraduates, mathematicians, and anyone wanting to get a quick overview of the subject of commutative harmonic analysis. Applications are to mathematical physics, engineering and other parts of hard science. Required background is calculus, set theory, integration theory and the theory of sequences and series.

CAR/27