Harmonic Analysis: from Fourier to Haar Maŕıa

Contents Introduction xv Chapter 1. Fourier series: some motivation 1 1.1. Some examples and key definitions 1 1.2. Main questions 5 1.3. Fourier series and Fourier coefficients 7 1.4. A little history, and motivation from the physical world 11 Chapter 2. Interlude 17 2.1. Nested classes of functions on bounded intervals 17 2.2. Modes of convergence 28 2.3. Interchanging limit operations 34 2.4. Density 38 Chapter 3. Pointwise convergence of Fourier series 41 3.1. Pointwise convergence: why do we care? 41 3.2. Smoothness vs convergence 44 3.3. A suite of convergence theorems 50 Chapter 4. Summability methods: Dirichlet kernels and Fejér kernels 55 4.1. Partial Fourier sums and the Dirichlet kernel D N 55 4.2. Properties of the Dirichlet kernel 57 4.3. Convolution 59 4.4. Good kernels, or approximations of the identity 67 4.5. Fejér kernels and Cesàro means 72 4.6. Poisson kernels and Abel means 75 4.7. Excursion into L p (T) 76 Chapter 5. Mean-square convergence of Fourier series 79 5.1. Basic Fourier theorems in L 2 (T) 79 5.2. Geometry of the Hilbert space L 2 (T) 81 5.3. Completeness of the trigonometric system 85 5.4. Equivalent conditions for completeness in L 2 (T) 90 Chapter 6. A whirlwind tour of discrete Fourier and Haar analysis 95 6.1. Fourier Series: a summary 95 6.2. The discrete Fourier basis 96 6.3. A parenthesis on bases and dual bases in C N 100 6.4. The discrete Fourier transform and its inverse 101 6.5. The Fast Fourier Transform (FFT) 102 6.6. The discrete Haar basis 107 v vi CONTENTS