Chap1 - Christopher Heil Introduction to Harmonic Analysis Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo Contents General

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Unformatted text preview: Christopher Heil Introduction to Harmonic Analysis November 12, 2010 Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo 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 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561....
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This note was uploaded on 08/25/2011 for the course MATH 7337 taught by Professor Staff during the Fall '08 term at Georgia Institute of Technology.

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Chap1 - Christopher Heil Introduction to Harmonic Analysis Springer Berlin Heidelberg NewYork HongKong London Milan Paris Tokyo Contents General

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