A Panorama of Harmonic Analysis

AMS / MAA THE CARUS MATHEMATICAL MONOGRAPHS VOL 27 A PANORAMA OF HARMONIC ANALYSIS 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. ⃝1 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 − D 0 @x2 @t2 u.x; 0/ D 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 C 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´ commu- tators, 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.

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