X CONTENTS 1.3.3 *Rates of Convergence in L2 39 1.3.3.1 Application to Absolutely-Convergent Fourier Series 43 1.4 Norm Convergence and SummabiHty 45 1.4.1 Approximate Identities 45 1.4.1.1 Almost-Everywhere Convergence of the Abel Means 49 1.4.2 Summability Matrices 51 1.4.3 Fejer Means of a Fourier Series 54 1.4.3.1 Wiener's Closure Theorem on the Circle 57 1.4.4 *Equidistribution Modulo One 57 1.4.5 *Hardy's Tauberian Theorem 59 1.5 Improved Trigonometric Approximation 61 1.5.1 Rates of Convergence in C(T) 61 1.5.2 Approximation with Fejer Means 62 1.5.3 *Jackson's Theorem 65 1.5.4 *Higher-Order Approximation 66 1.5.5 ^Converse Theorems of Bernstein 70 1.6 Divergence of Fourier Series 73 1.6.1 The Example of du Bois-Reymond 74 1.6.2 Analysis via Lebesgue Constants 75 1.6.3 Divergence in the Space L1 78 1.7 * Appendix: Complements on Laplace's Method 80 1.7.0.1 First Variation on the Theme-Gaussian Approximation 80 1.7.0.2 Second Variation on the Theme-Improved Error Estimate 80 1.7.1 * Application to Bessel Functions 81 1.7.2 *The Local Limit Theorem of DeMoivre-Laplace 82 1.8 Appendix: Proof of the Uniform Boundedness Theorem 84 1.9 * Appendix: Higher-Order Bessel functions 85 1.10 Appendix: Cantor's Uniqueness Theorem 86 2 FOURIER TRANSFORMS ON THE LINE AND SPACE 89 2.1 Motivation and Heuristics 89 2.2 Basic Properties of the Fourier Transform 91 2.2.1 Riemann-Lebesgue Lemma 94 2.2.2 Approximate Identities and Gaussian Summability 97 2.2.2.1 Improved Approximate Identities for Pointwise Convergence 100 2.2.2.2 Application to the Fourier Transform 102 2.2.2.3 The ^-Dimensional Poisson Kernel 106
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