Cbms-nsf regional conference series in applied mathematics
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( x ) ⇔ decay of Fourier coefficients f k . . . . . 169 B.5 Relations with shifts and derivatives . . . . . . . . . . . . . . . . . 170 B.6 Product-convolution relations . . . . . . . . . . . . . . . . . . . . . 171 B.7 Convolution on n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 B.8 Fourier transform (FT) . . . . . . . . . . . . . . . . . . . . . . . . . . 172 B.9 Properties of the FT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 B.10 FT of some common functions . . . . . . . . . . . . . . . . . . . . 174 B.11 The Paley–Wiener theorem . . . . . . . . . . . . . . . . . . . . . . . 174 B.12 Smoothness and decay of the FT . . . . . . . . . . . . . . . . . . . . 175 B.13 Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B.14 Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Downloaded 11/25/23 to 84.54.70.76 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy xii Contents B.15 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 B.16 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 B.17 Wavefront sets of distributions . . . . . . . . . . . . . . . . . . . . . 180 C Geometric Rays, Trapping, Energy Decay 181 C.1 Geometric rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 C.2 Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 C.3 Local energy decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 D Some Classes of Linear Operators and Operator Functions 183 D.1 Some notation concerning linear operators . . . . . . . . . . . . . 183 D.2 Fredholm and semi-Fredholm operators. Fredholm index. . . . 183 D.3 Schatten–von Neumann classes . . . . . . . . . . . . . . . . . . . . . 185 D.4 Regularized determinants . . . . . . . . . . . . . . . . . . . . . . . . 187 D.5 Analytic Fredholm operator functions . . . . . . . . . . . . . . . . 188 E Recommended Books and Surveys 191 Bibliography 193 Index 237 Downloaded 11/25/23 to 84.54.70.76 . Redistribution subject to SIAM license or copyright; see https://epubs.siam.org/terms-privacy |