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Harmonic Analysis Is a Broad field Involving a Great Deal of Subjects Concerning the Art of Decomposing Functions Into Constituent Parts

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Harmonic Analysis Is a Broad field Involving a Great Deal of Subjects Concerning the Art of Decomposing Functions Into Constituent Parts Lecture Notes in Harmonic Analysis Lectures by Dr. Charles Moore Throughout this document, signifies end proof, and N signifies end of example. Table of Contents Table of Contents i Lecture 1 Introduction to Fourier Analysis 1 1.1 Fourier Analysis . 1 1.2 In more general settings. 3 Lecture 2 More Fourier Analysis 4 2.1 Elementary Facts from Fourier Analysis . 4 Lecture 3 Convolving Functions 7 3.1 Properties of Convolution . 7 Lecture 4 An Application 9 4.1 Photographing a Star . 9 4.2 Results in L2 ............................. 10 Lecture 5 Hilbert Spaces 12 5.1 Fourier Series on L2 ......................... 12 Lecture 6 More on Hilbert Spaces 15 6.1 Haar Functions . 15 6.2 Fourier Transform on L2 ....................... 16 Lecture 7 Inverse Fourier Transform 17 7.1 Undoing Fourier Transforms . 17 Lecture 8 Fejer Kernels 19 8.1 Fejer Kernels and Approximate Identities . 19 Lecture 9 Convergence of Cesar´oMeans 22 9.1 Convergence of Fourier Sums . 22 Notes by Jakob Streipel. Last updated December 1, 2017. i TABLE OF CONTENTS ii Lecture 10 Toward Convergance of Partial Sums 24 10.1 Dirichlet Kernels . 24 10.2 Convergence for Continuous Functions . 25 Lecture 11 Convergence in Lp 26 11.1 Convergence in Lp .......................... 26 11.2 Almost Everywhere Convergence . 28 Lecture 12 Maximal Functions 29 12.1 Hardy-Littlewood Maximal Functions . 29 Lecture 13 More on Maximal Functions 33 13.1 Proof of Hardy-Littlewood's Theorem . 33 Lecture 14 Marcinkiewicz Interpolation 36 14.1 Proof of Marcinkiewicz Interpolation Theorem . 36 Lecture 15 Lebesgue Differentiation Theorem 38 15.1 A Note About Maximal Functions . 38 15.2 Lebesgue Differentiation Theorem . 39 Lecture 16 Maximal Functions and Kernels 41 16.1 Generalising Lebesgue Differentiation Theorem . 41 Lecture 17 Rising Sun Lemma 44 17.1 Nontangential Maximal Function . 44 17.2 Riesz's Proof of the Hardy-Littlewood Theorem . 46 Lecture 18 Calder´on-Zygmund Decomposition of Functions 47 18.1 Higher-Dimensional Rising Sun Lemma . 47 Lecture 19 Density of Sets 49 19.1 Hardy-Littlewood's Theorem from Calder´on-Zygmund . 49 19.2 Density of Sets . 50 Lecture 20 Marcinkiewicz Integral 52 20.1 Convergence of Marcinkiewicz Integral . 52 Lecture 21 Integral Operators 54 21.1 Schur's Lemma . 54 Lecture 22 Integral Operators continued 57 22.1 Singular Integrals . 57 Lecture 23 Integral Operators continued 60 23.1 Finishing the Proof . 60 Lecture 24 Integral Operators continued 63 24.1 Proof of the Lemma . 63 Lecture 25 Integral Operators continued 64 25.1 Finalising the Proof . 64 TABLE OF CONTENTS iii Index 67 INTRODUCTION TO FOURIER ANALYSIS 1 Lecture 1 Introduction to Fourier Analysis Harmonic analysis is a broad field involving a great deal of subjects concerning the art of decomposing functions into constituent parts. These might be Fourier coefficients, breaking them down into exponential parts, wavelet theory, tools to deal with partial differential equations, or Sobolev spaces. This course will deal with the following: • Fourier analysis, • Harmonic functions, • Singular integrals, and • Maximal functions. 1.1 Fourier Analysis Definition 1.1.1 (Inner product space). Let V be a finite dimensional vector space over C. Then V is called an inner product space if there is a mapping h·; ·i: V × V ! C which satisfies the following for all vectors x; y; z 2 V and scalars α 2 C: (i) Conjugate symmetry, meaning that hx; yi = hy; xi; (ii) Linearity in the first argument, i.e. hαx; yi = αhx; yi and hx + y; zi = hx; zi + hy; zi; (iii) Positive-definiteness, meaning that hx; xi ≥ 0 and hx; xi = 0 if and only if x = 0. All inner product spaces automatically induce a norm, namely kvk = hv; vi1=2. Moreover normed spaces are automatically metric spaces by d(v; w) = kv − wk. Ergo it has the following properties: Definition 1.1.2 (Metric). A function d(·; ·): V × V ! R is called a metric if (i) d(x; y) ≥ 0 and d(x; y) = 0 if and only if x = y; (ii) symmetry, i.e. d(x; y) = d(y; x); and (iii) d(x; z) ≤ d(x; y) + d(y; z) for all x; y; x 2 V . Since we have an inner product, we are able to define all manner of other interesting concepts. Definition 1.1.3 (Orthogonal, orthonormal). A basis B = f v1; v2;:::; vn g is called orthogonal if the basis elements are pairwise orthogonal, i.e. hvi; vji = 0 for all i 6= j. Moreover the basis is called orthonormal if in addition kvik = 1 for all i. INTRODUCTION TO FOURIER ANALYSIS 2 Since a basis spans the space, we can write v = c1v1 + c2v2 + ::: + cnvn. Now if the basis in addition is orthonormal we have the illuminating property that hv; v1i = hc1v1 + c2v2 + ::: + cnvn; v1i = c1hv1; v1i + c2hv1; v2i + ::: + cnhv1; vni 2 = c1kv1k + 0 + 0 + ::: + 0 = c1; and similarly hv; vii = ci for all i. Therefore v = hv; v1iv1 + hv; v2iv2 + ::: + hv; vnivn; and hvvv; vii are called the Fourier coefficients of v. We can do the same thing on a slightly more interesting space than ordinary Euclidean space, namely L1: Definition 1.1.4. Suppose f is a function on [−π; π) with Z πjf(x)j dx < 1 −π meaning that f 2 L1[−π; π). Then for n 2 Z we define 1 Z π f^(n) = f(x)e−inx dx; 2π −π called the nth Fourier coefficient of f. On [−π; π), consider the set f einx g . Define n2Z 1 Z π hf; gi = f(x)g(x) dx: 2π −π Exercise 1.1.5. Show that this is indeed an inner product. Solution. First conjugate symmetry, and clearly 1 Z π 1 Z π hf; gi = f(x)g(x) dx = f(x)g(x) dx 2π −π 2π −π 1 Z π 1 Z π = f(x)g(x) dx = g(x)f(x) dx = hg; fi: 2π −π 2π −π Linearity follows by integration being linear on its own, and note for positive- definiteness that 1 Z π 1 Z π hf; fi = f(x)f(x) dx = jf(x)j2 dx ≥ 0 2π −π 2π −π since for any complex number z = a + ib, zz = (a + ib)(a − ib) = a2 − i2b2 = a2 + b2 = jzj2: For the final condition, note that in fact they are implied by the first three for all inner products. To see this, take f = 0 in linearity, i.e. h0; 0i = hg; 0i + h−g; 0i = hg; 0i − hg; 0i = 0 and from positive-definiteness we have that hx; xi = 0 implies x = 0. INTRODUCTION TO FOURIER ANALYSIS 3 Now suppose that n 6= m are integers. Then 1 Z π 1 Z π heinx; eimxi = einxeimx dx = einxe−imx dx 2π −π 2π −π Z π i(n−m)x π 1 i(n−m)x 1 e = e dx = 2π −π 2π i(n − m) x=−π 1 ei(n−m)π e−i(n−m)π = − = 0; 2π i(n − m) i(n − m) and 1 Z π 1=2 1 Z π 1=2 keinxk = heinx; einxi = einxeinx dx = e0 = 1: 2π −π 2π −π Therefore f einx g is an orthonormal set. n2Z Given a function f, we defined f^(n) = hf; einxi. We would like for this to mean that X f = f^(n)einx n2Z like v = hv; v1iv1 + hv; v2iv2 + ::: + hv; vnivn: But is it a basis? In what sense does this infinite sum converge? 1.2 In more general settings. R π ^ inx Given f on [−π; π) and −πjf(x)j dx < 1 we defined f(n) = hf; e i. Then P f^(n)einx is called the Fourier series of f, often written n2Z X f ∼ f^(n)einx: n2Z Now it is extremely important to note that this means only that f^(n) is defined as the integral we discussed previously, and nothing more. R π inx In general, if µ is a measure on [−π; π), thenµ ^(n) = −π e dµ(x), and indeed if f is a function than f(x) dx is an example of a measure. We can do it more generally in higher dimensions. If f : Rn ! C, and if R n jf(x)j dx < 1, we define R Z f^(x) = eix·tf(t) dt; n R where x = (x1; x2; : : : ; xn), t = (t1; t2; : : : ; tn), and x·t = x1t1+x2t2+:::+xntn. This f^: Rn ! C is called the Fourier transform. Remark 1.2.1. This is of course not unlike, say, the Laplace transform, Z 1 Lffg(s) = f(t)e−st dt: 0 MORE FOURIER ANALYSIS 4 Indeed we could consider n-dimensional tori too, if we like, where'd we'd be doing the exact same thing, just over a different integrating domain (and with a different scalar in front). More generally we may do this over any locally compact Abelian group G (which is what it sounds like, with the caveat that the group structure and the topological structure be connected in that the group operations of addition and negation are both continuous). It is a fact that on G there exists a measure µ called the Haar measure such that it is translation invariant, i.e. µ(E + x) = µ(E), for evert Borel set E ⊆ G and x 2 G. If γ : G ! C is a mapping such that γ(x + y) = γ(x)γ(y) for every x; y 2 G, e.g. γ(x) = einx, then γ is called a (multiplicative) character on G. In L2[−π; π), einx are characters.
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