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chap1 - Christopher Heil Introduction to Harmonic Analysis...

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Christopher Heil Introduction to Harmonic Analysis November 12, 2010 Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo
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Contents General Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1 The Fourier Transform on L 1 ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Fourier Transform on L 1 ( R ) . . . . . . . . . . . . . . . . . . . . 1 1.1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Translation, Modulation, Dilation, and Involution . . . . . . . . . . . 10 1.2.1 Four Fundamental Operators . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 The Riemann–Lebesgue Lemma . . . . . . . . . . . . . . . . . . . . . 13 1.2.3 Position and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.4 The HRT Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.1 Some Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . 17 1.3.2 Definition and Basic Properties of Convolution . . . . . . . . 18 1.3.3 Young’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.3.4 Convolution as Filtering; Lack of an Identity . . . . . . . . . . 21 1.3.5 Convolution as Averaging; Introduction to Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.6 Convolution as an Inner Product . . . . . . . . . . . . . . . . . . . . 25 1.3.7 Convolution and Smoothing . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.8 Convolution and Differentiation . . . . . . . . . . . . . . . . . . . . . 28 1.3.9 Convolution and Banach Algebras . . . . . . . . . . . . . . . . . . . 29 1.3.10 Convolution on General Domains . . . . . . . . . . . . . . . . . . . . 32 1.4 The Duality Between Smoothness and Decay . . . . . . . . . . . . . . . . 36 1.4.1 Decay in Time Implies Smoothness in Frequency . . . . . . 36 1.4.2 A Primer on Absolute Continuity . . . . . . . . . . . . . . . . . . . . 38 1.4.3 Smoothness in Time Implies Decay in Frequency . . . . . . 41 1.4.4 The Riemann–Lebesgue Lemma Revisited . . . . . . . . . . . . 42 1.5 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.5.1 Definition and Existence of Approximate Identities . . . . 46 1.5.2 Approximation in L p ( R ) by an Approximate Identity . . 48
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vi Contents 1.5.3 Uniform Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 1.5.4 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 1.5.5 Dense Sets of Nice Functions . . . . . . . . . . . . . . . . . . . . . . . . 54 1.5.6 The C Urysohn Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.5.7 Gibbs’s Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 1.5.8 Translation-Invariant Subspaces of L 1 ( R ) . . . . . . . . . . . . . 57 1.6 The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.6.1 The Fej´ er Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 1.6.2 Proof of the Inversion Formula . . . . . . . . . . . . . . . . . . . . . . 62 1.6.3 Summability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 1.6.4 Pointwise Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.6.5 Decay and Smoothness Revisited . . . . . . . . . . . . . . . . . . . . 69 1.7 The Range of the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . 72 1.8 Some Special Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 1.9 The Schwartz Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1.9.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . 80 1.9.2 Topology and Convergence in the Schwartz Space . . . . . 80 1.9.3 Invariance of the Schwartz Space . . . . . . . . . . . . . . . . . . . . 81 2 Fourier Series and the Abstract Fourier Transform . . . . . . . . 85 2.1 The Abstract Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.1.1 Examples of Locally Compact Abelian Groups . . . . . . . . 85 2.1.2 Characters and the Fourier Transform . . . . . . . . . . . . . . . 87 2.1.3 The Fourier Transform on LCA Groups . . . . . . . . . . . . . . 90 2.2 Fourier Series and Approximate Identities on the Torus . . . . . . 92 2.2.1 Partial Sums and the Dirichlet and Fej´ er Kernels . . . . . . 96 2.2.2 Approximate Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.2.3 Ces` aro Summability and the Inversion Formula . . . . . . . 101 2.3 Completeness and the L 2 -Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.3.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 2.3.2 The L 2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.3.3 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.4 Weyl’s Equidistribution Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.5 Basis Properties of the Trignometric System . . . . . . . . . . . . . . . . 115 2.5.1 Schauder Bases and the Partial Sum Operators . . . . . . . 116 2.5.2 The Symmetric Partial Sum Operators and Their Relatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.6 The Conjugate Function and Norm Convergence of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 2.7 Pointwise Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.8 The Poisson Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.9 Wiener’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.10 Wiener’s Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 2.11 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
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Contents vii 3 The Fourier Transform on L 2 ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.1 Definition and Basic Properties of the Fourier Transform on L 2 ( R ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 3.2 The Hermite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.2.1 Construction of the Hermite Functions . . . . . . . . . . . . . . . 156 3.2.2
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