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A Course in Harmonic Analysis

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A Course in Harmonic Analysis A Course in Harmonic Analysis Jason Murphy Missouri University of Science and Technology Contents 1 Introduction 4 2 Fourier analysis, part I 7 2.1 Separation of variables . .7 2.2 Fourier series in general . .9 2.3 Fourier series, revisited . 12 2.4 Convergence of Fourier series . 14 2.5 The Fourier transform . 20 2.5.1 Remarks about pointwise convergence . 26 2.6 Applications to PDE . 28 2.7 The Fourier transform of distributions . 30 2.8 The Paley{Wiener theorem . 32 2.9 Exercises . 34 3 Fourier analysis, part II 38 3.1 Sampling of signals . 38 3.2 Discrete Fourier transform . 43 3.3 Fast Fourier transform . 48 3.4 Compressed sensing . 53 3.5 Exercises . 72 4 Abstract Fourier analysis 74 4.1 Preliminaries . 74 4.2 Locally compact abelian groups . 77 4.3 Compact groups . 83 4.4 Examples . 86 4.5 Exercises . 96 2 CONTENTS 3 5 Wavelet transforms 97 5.1 Continuous wavelet transforms . 97 5.2 Discrete wavelet transforms . 110 5.3 Multiresolution analysis . 121 5.4 Exercises . 128 6 Classical harmonic analysis, part I 129 6.1 Interpolation . 129 6.2 Some classical inequalities . 139 6.3 Hardy{Littlewood maximal function . 144 6.4 Calder´on{Zygmund theory . 153 6.5 Exercises . 162 7 Classical harmonic analysis, part II 163 7.1 Mihlin multiplier theorem . 163 7.2 Littlewood{Paley theory . 166 7.3 Oscillatory integrals . 179 7.4 Exercises . 193 8 Modern harmonic analysis, part I 196 8.1 Semiclassical analysis . 196 8.2 Coifman{Meyer multipliers . 213 8.3 Exercises . 219 9 Modern harmonic analysis, part II 220 9.1 Sharp Gagliardo{Nirenberg . 220 9.2 Sharp Sobolev embedding . 230 9.3 Exercises . 237 10 Modern harmonic analysis, part III 238 10.1 Restriction theory . 238 10.2 Strichartz estimates . 244 10.3 More restriction theory . 251 10.4 Exercises . 255 A Prerequisite material 256 A.1 Lebesgue spaces . 256 A.2 Hilbert spaces . 260 A.3 Analysis tools . 266 A.4 Exercises . 271 Chapter 1 Introduction These notes were written to accompany the courses Math 6461 and Math 6462 (Harmonic Analysis I and II) at Missouri University of Science & Tech- nology. The goal of these notes is to provide an introduction into a range of top- ics and techniques in harmonic analysis, covering material that is interesting not only to students of pure mathematics, but also to those interested in applications in computer science, engineering, physics, and so on. We will focus on giving an overall sense of the available results and the analytic tech- niques used to prove them; in particular, complete generality or completely optimal results may not always be pursued. Technical details will sometimes be left to the reader to work out as exercises; solving these exercises is an important part of solidifying the reader's understanding of the material. At times we will not develop the full theory but rather give a survey of results, along with citations to references containing full details. These notes are organized as follows: • In Chapter 2, we introduce Fourier series, motivating their develop- ment through an application to solving PDE (a common theme for us). We then develop the Fourier transform, also providing some ap- plications to PDE. Other topics are discussed, including questions of pointwise convergence, the Fourier transform on distributions, and the Paley{Wiener theorem. • In Chapter 3 we discuss the question of sampling of signals (e.g. the Shannon{Nyquist theorem), as well as the discrete and fast Fourier transform. We close this chapter with a discussion of compressed sens- ing, providing a relatively complete presentation of the result of Can- 4 5 des, Romberg, and Tao [4] on reconstruction of signals using randomly sampled Fourier coefficients. • In Chapter 4, we present a survey of results in abstract Fourier trans- form, relying primarily on the textbook of Folland [9]. In particular, we demonstrate how many of the preceding topics may be viewed un- der the same umbrella (i.e. Fourier analysis on locally compact abelian groups). Most results are stated without proof. We then briefly discuss the case of Fourier analysis on compact groups and present a few im- portant examples in detail (namely, SU(2) and SO(n) for n 2 f3; 4g). • In Chapter 5, we discuss the continuous and discrete wavelet trans- forms, as well as the notion of multiresolution analysis. In addition to wavelet transforms, we also frequently discuss the `windowed' Fourier transform. Our primary reference is the book of Daubechies [7]. This chapter provides a relatively brief introduction into a very rich topic with a wide range of applications. • In Chapter 6, we begin discussing what I have called `classical' har- monic analysis (although this distinction of `classical' versus `modern' should not be taken too seriously). This includes the theory of interpo- lation of linear operators, some `classical inequalities' (like convolution inequalities and Sobolev embedding), the Hardy{Littlewood maximal function (and vector maximal function), and finally the Calder´on{ Zygmund theory for singular integral operators. • In Chapter 7, we continue the study of `classical' topics in harmonic analysis. We firstly prove the Mihlin multiplier theorem. We then de- velop Littlewood{Paley theory (including the Littlewood{Paley square function estimate and some fractional calculus estimates). Finally, we study oscillatory integrals (proving, for example, the stationary phase theorem and providing some applications to PDE). • In Chapter 8, we begin our study of more `modern' topics in harmonic analysis. We begin with a study of semiclassical analysis. This is not actually a subfield of harmonic analysis; however, it is closely related due to the frequent analysis of oscillatory integrals. We give a brief introduction based on the textbook of Martinez [19]; we get as far as the proof of L2 boundedness for pseudodifferential operators. In the rest of this chapter, we prove the non-endpoint cases of the Coifman{ Meyer multiplier theorem. 6 CHAPTER 1. INTRODUCTION • In Chapter 9, we continue our study of `modern' topics and turn to the question of sharp inequalities and existence of optimizers. We consider two examples, namely, the Gagliardo{Nirenberg inequality and Sobolev embedding. For Gagliardo{Nirenberg, we present a proof based on radial decreasing rearrangements and the compactness of the radial Sobolev embedding. For Sobolev embedding, we present a proof based on profile decompositions, thus giving a short introduction into `concentration-compactness' techniques (which have come to play an important role in the setting of nonlinear PDE). • In Chapter 10, we prove some basic results in `restriction theory'. This refers to the question of when it makes sense to restrict a function's Fourier transform to a surface. We begin with a result due to Strichartz for the paraboloid. This result can be interpreted as a space-time estimate for solutions to the linear Schr¨odingerequation. We take a slight detour to prove a wider range of such estimates (which now go by the name of Strichartz estimates). We then return to restriction theory and prove the `Tomas{Stein' result (up to the endpoint) for the case of the sphere. • Finally, in the appendix, we have collected some prerequisite material for the reader's reference. The material from these notes has been drawn from many different sources. In addition to the references listed in the bibliography, this in- cludes the author's personal notes from a harmonic analysis course given by M. Visan at UCLA. The author gratefully acknowledges support from the University of Mis- souri Affordable and Open Educational Resources Initiative Award, as well as the students of Math 6461-6462 for their useful feedback and corrections. Chapter 2 Fourier analysis, part I 2.1 Separation of variables Consider the following partial differential equation (PDE): 8 @ u = @2u (t; x) 2 (0; 1) × (0; 1) <> t x u(0; x) = f(x) x 2 (0; 1) (2.1) :>u(t; 0) = u(t; 1) = 0 t 2 [0; 1); where f : (0; 1) ! R is some given function. This is the well-known heat equation. This is an example of an initial-value problem (the solution is specified at t = 0), as well as a boundary-value problem (the values of the solution are prescribed at the boundary of the spatial domain (0; 1)). One approach to solving PDEs like this is the method of separation of variables, which entails looking for separated solutions of the form u(t; x) = p(t)q(x): Using (2.1) and rearranging, we find that for u to be a solution we must have −p0(t) −q00(x) = : p(t) q(x) As the left-hand side depends only on t and the right-hand side depends only on x, we are led to the problem p0(t) = −λp(t) and − q00(x) = λq(x) for some constant λ. The equation for p is solvable for any λ; indeed, p(t) = e−λtp(0) does the job. The problem for q is more interesting, since in addition to the ordinary 7 8 CHAPTER 2. FOURIER ANALYSIS, PART I differential equation (ODE) it must also satisfy the boundary conditions. One finds that there are solutions only for special choices of λ, namely, λ = (nπ)2 for some integer n > 0. A corresponding solution is then given by q(x) = sin(nπx). What we have therefore discovered is that the method of separation of variables yields a countable family of solutions to the heat equation satisfying the boundary conditions, namely −(nπ)2t e sin(nπx)cn for any n > 0 and cn 2 R: Furthermore, any linear combination of these solutions solves the PDE and satisfies the boundary condition.
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