CONTENTS XI 2.2.3 Fourier Transforms of Tempered Distributions 108 2.2.4 ^Characterization of the Gaussian Density 109 2.2.5 *Wiener's Density Theorem 110 2.3 Fourier Inversion in One Dimension 112 2.3.1 Dirichlet Kernel and Symmetric Partial Sums 112 2.3.2 Example of the Indicator Function 114 2.3.3 Gibbs-Wilbraham Phenomenon 115 2.3.4 Dini Convergence Theorem 115 2.3.4.1 Extension to Fourier's Single Integral 117 2.3.5 Smoothing Operations in I^-Averaging and Summability 117 2.3.6 Averaging and Weak Convergence 118 2.3.7 Cesaro Summability 119 2.3.7.1 Approximation Properties of the Fejer Kernel 121 2.3.8 Bernstein's Inequality 122 2.3.9 * One-Sided Fourier Integral Representation 124 2.3.9.1 Fourier Cosine Transform 124 2.3.9.2 Fourier Sine Transform 125 2.3.9.3 Generalized /i-Transform 125 2.4 L2 Theory in W1 128 2.4.1 Plancherel's Theorem 128 2.4.2 *Bernstein's Theorem for Fourier Transforms 129 2.4.3 The Uncertainty Principle 131 2.4.3.1 Uncertainty Principle on the Circle 133 2.4.4 Spectral Analysis of the Fourier Transform 134 2.4.4.1 Hermite Polynomials 134 2.4.4.2 Eigenfunction of the Fourier Transform 136 2.4.4.3 Orthogonality Properties 137 2.4.4.4 Completeness 138 2.5 Spherical Fourier Inversion in En 139 2.5.1 Bochner's Approach 139 2.5.2 Piecewise Smooth Viewpoint 145 2.5.3 Relations with the Wave Equation 146 2.5.3.1 The Method of Brandolini and Colzani 149 2.5.4 Bochner-Riesz Summability 152 2.5.4.1 A General Theorem on Almost-Everywhere Summability 153 2.6 Bessel Functions 154 2.6.1 Fourier Transforms of Radial Functions 157 2.6.2 L2-Restriction Theorems for the Fourier Transform 158 2.6.2.1 An Improved Result 159 2.6.2.2 Limitations on the Range of p 161 2.7 The Method of Stationary Phase 162 2.7.1 Statement of the Result 163 2.7.2 Application to Bessel Functions 164 2.7.3 Proof of the Method of Stationary Phase 165 2.7.4 Abel's Lemma 167
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