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Index algorithm, 253, 255, 261, 271 aliasing error, 5, 63 astronomy, 6, 26, 250, 369

basis Riesz, 56-58, 60, 61, 68, 70,

72, 134, 135, 139, 144, 198, 218

Schauder, 15 unconditional, 15, 191

Bayesian estimation, 26, 357 Benedetto, 10 Berenstein, 199 Besov space, 23, 81, 180, 181, 184,

187 Beurling, 11, 13, 20, 168, 169 Beutler, 11 Bezout equation, 24,196,201,203,

211, 212, 215 Birkhoff, 12 Borel, 1 Brillouin zone, 6 Butzer, 4, 24, 168, 232

cardinal function, 2 Carleman, 12 Cauchy, 1 Cauchy's sampling theorem, 1 classical sampling theorem, 3, 49,

141, 232, 273, 319 Cohen, 9 complete, 10, 16, 18, 198, 212, 217,

218 congruence, 7 cotabular functions, 2 crystallography, 6

Dai, 10

de Branges, 108 de Bruijn, 74 de la Vallee-Poussin, 1, 2 deconvolution, 23, 200, 219 deconvolvers, 195, 203 density, 11

lower, 13 natural, 12 P6lya, 14 uniform, 12, 13, 19 upper, 13

Dini, 12, 15 distribution, 107, 134 Doppler imaging, 26, 369 Dubois, 7 Duffin, 11, 15, 19 Duijndam, 257 Dutt, 251, 256

Eachus, 11 eigenfunction, 224 eigenspace, 224 eigenvalue

multiplicity, 224 simple, 224

embedding theorems, 22 error

aliasing, 63 Escher, 10 Euler, 4

fast computation Bessel transform, 271 NDFT, 253, 255, 261

Feichtinger, 20 FFT, 25, 26, 163, 249, 250, 252

255, 261-264 filter bank, 25, 281

fractional Fourier transform, 90 frame, 5, 15, 62, 134, 135, 198,

218 chirplet, 97 coefficient, 95 exact, 15, 20, 21 Fourier, 18 Gabor, 5, 198 metaplectic, 97 operator, 15, 308 overcomplete, 21 tight, 15, 113, 135

frame bounds, 15, 16, 158, 218 Franklin wavelet, 62

Gabor frame, 5, 198 Gabor system, 5, 16 Gaussian quadrature, 8 Grochenig, 20 Gram operator, 17, 60 group

Heisenberg, 95 LCA, 4,7,8 metaplectic, 74, 95

Hormander, 196, 198, 200, 201 Haas, 9 Hadamard, 1,2 Heisenberg group, 95 Hermite interpolation, 67, 68 Higgins, 4, 7 Hinsen, 168

interpolation Hermite, 22, 67, 68 Jacobi, 201, 203, 205 Lagrange, 1, 203, 251 Newton-Gauss, 2

irregular sampling, see nonunifor­m sampling

iteration, 17, 285, 357

Jacobi interpolation, 201, 203, 205 Jaffard, 19 Jerri, 4

Kahane, 11, 14, 18 Katsnelson, 21 Kelly, 149 Kon, 149

Index 415

Koosis, 11, 14 Kotel'nikov, 4, 21, 49 Kramer, 222 Kramer's lemma, 24, 221, 222, 226,

227 Krein, 22, 107

Lagarias, 9 Lagrange interpolation, 1, 203, 251 Landau, 11, 19 Larson, 10 lattice, 6, 374 LCA group, 4, 7, 8 Leon, 10 Leonardo da Vinci, 10 Levinson, 11, 14 Littlewood, 7 Littlewood-Paley theory, 81 local deconvolution, 196 lower frame bound, 218

Maclaurin, 4 Madych,9 Malliavin, 11 Meisters, 198 Mersereau, 7 metaplectic group, 74, 95 Meyer wavelet, 64 Middleton, 6, 374 minimal, 217, 218 Miyakawa,6 modulation spaces, 22 MRI, 20, 26, 347 multiband, 25, 273, 275 multidimensional sampling theo-

rem, 6 multi resolution , 9, 22, 23 multisensor deconvolution, 195 multiwavelet, 22, 67, 72

Newton-Gauss interpolation, 2

416 Index

non-commensurate, 203 non-uniform sampling, 10, 320 nonperiodic frame, 217 nonperiodic set, 212, 217 Nyquist rate, 3, 11, 276

P6lya, 7, 12 Paley, 10 Paley-Wiener space, 3, 50, 108,

176, 190 Pelt, 257 periodization, 4 Petersen, 6, 198, 374 POCS, 25, 285 Poincare, 2 point spectrum, 224 pointwise convergence, 21, 23 Poisson integral, 130 Poisson summation formula, 4, 22,

75 Pompeiu problem, 199 positive-definite extensions, 22, 107 potential theory, 14 prime number theorem, 1 Prosser, 6 pseudo-inverse, 17, 158

QMF, 9 quantization error, 26 quasi-analytic functions, 12

Radon transform, 26, 373, 375 Raphael, 149 Redheffer, 11, 14 regular sampling, see uniform sam­

pling remote sensors, 200 reproducing kernel Hilbert spaces,

see RKHS Riemann, 2 Riemann zeta function, 2 Riesz, 50 Riesz basis, 56-58, 60, 61, 68, 70,

72, 134, 135, 139, 144, 198, 218

RKHS, 49-51 Rogosinsky means, 24 Rokhlin, 251, 256

sampling hexagonal, 7 minimum rate, 275 multidimensional, 5, 6, 23, 26,

158 non-uniform, 10, 19, 320 nonperiodic, 212

sampling function, 49 sampling multiplier, 4 sampling rate, 3 sampling theorem

Cauchy's, 1 classical, 3, 49, 141, 232, 273,

319 Kluvanek's, 8 multidimensional, 6 non-uniform, 19 nonbandlimited signals, 5

scaling function, 9, 49 Schaeffer, 11, 15, 19 Schauder basis, 15 Schoneville, 257 Selberg, 4 separated, 10, 20 set of uniqueness, 212, 215, 217 Shannon, 3 sinc, 232 sinc function, 3, 49 Soardi, 10 Sobolev space, 22, 53, 54, 109, 181 Steffensen, 2 stochastic processes, 6 Strohmer, 168 strongly coprime, 24, 196, 198, 200,

203, 204 Sturm-Liouville problem, 224 Szasz, 12

Tauberian theorem, 2 Thomas, 21 tiling, 6, 8

Toeplitz matrix, 161, 163, 300-302

block, 161, 163,250,302,308 tomography, 26, 347, 373 trace formulas, 4

uncertainty principle, 22, 73 unconditional basis, 15, 191 uniform sampling, 3 uniformly discrete, see separated upper frame bound, 218

visibility, 17 Vitali, 18 von Koch, 1 von Neumann, 5 Voronoi cell, 6, 168, 355

Walsh, 12 Walter, 149 Wang, 9

Index 417

wavelet, 5, 198 auditory modeling, 16 Franklin, 62 Meyer, 64 Shannon, 8, 10, 23

weighted norm inequalities, 22 Weiland, 10 Whittaker, E. T., 2, 4 Whittaker, J. M., 2, 7, 49 Wiener, 2, 10, 12 Wigner distribution, 90, 97

Yao, 21, 50 Yger, 199 Young, 12, 218

Zakharov, 10 Zayed,4 zeta function, 2 Zygmund, 187

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L. Debnath: Wavelet Transforms and Time-Frequency Signal Analysis (ISBN 0-8176-4104-1)

K. Grochenig: Foundations of Time-Frequency Analysis (ISBN 0-8176-4022-3)

D. Walnut: An Introduction to Wavelet Analysis (ISBN 0-8176-3962-4)