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Index

The keywords cite the main references only.

accelerating reference frame, 212, 244 acoustic wavelets, 273

absorbed, 278 emitted, 278

adjoint, 13, 24-25 admissibility condition, 67 admissible, 264 affine

group, 205, 257 transformations, 257

aliasing operator, 162 almost everywhere (a.e.), 22 ambiguity function, 250, 254

conformal cross, 266 self, 266

narrow-band, 254 cross, 254 self, 254

wide-band, 248 cross, 250 self, 250

analytic signal, 68, 206, 218, 219 lower 73 upper 73

analyzing operator, 83 antilinear, 12 atomic composition, 233 averaging

operator, 142 property, 141

Balian-Low theorem, 116 band-limited, 99 bandwidth, 99, 252 basis, 4

dual, 9 reciprocal, 16

Bernstein's inequality, 101

biorthogonal, 15 blurring, 144, 242 bounded, 26

linear functional, 27 bra, 14

carrier frequency, 252 causal, 205

cone, 222 tube, 206, 223

characteristic function, 40 chirp, 47 complete, 4, 80 complex

space-time, 206 structure, 207

conformal cross-ambiguity function, 250, 254,

266 group, 204 self-ambiguity function, 266 transformations, 260

special, 260, 269 consistency condition, 57, 87, 234, 249 counting measure, 40, 106 cross-ambiguity function, 250, 254 cumulants, 191-196

de-aliasing operator, 163, 164 dedicated wavelets, 204 demodulation, 253 detail, 63 diadic

interpolation, 189 rationals, 188, 190

differencing operator, 152 dilation equation, 142 Dirac notation, 14 distribution, 26

298 Index

Doppler effect, 243, 247 shift, 252

down-sampling operator, 149 duality relation, 9 duration, 206, 242

electromagnetic wavelets, 224 elementary

current, 279 scatterers, 263 scattering event, 265

Euclidean region, 225 evaluation map, 224 exact scaling functions, 191

filter, coefficients, 142 extremal-phase, 181 linear-phase, 181 maximum-phase, 181 minimum-phase, 181

finite impulse response (FIR), 155, focal point, 205 Fourier

coefficients, 27 series, 26 transform, 29

fractal, 183 frame, 82

bounds, 82 condition, 82 deformations of, 117-218 discrete 82, 91 generalized, 82 normalized 83 operator, 83 reciprocal, 86 snug 90 tight 82 vectors, 82

future cone, 222

Gabor expansions, 114 gauge freedom, 212

Haar wavelet, 159 harmonic oscillator, 118 helicity, 206, 210

high-pass filter, 147 sequence, 154

Hilbert space, 23 transform, 251, 252

Holder continuous, 187-188 exponents, 188

hyperbolic lattice, 124

ideal band-pass filter, 167 ideal low-pass filter, 110 identity operator, 10 image, 6 incomplete, 4 inner product, 12

standard, 12 instantaneous frequency, 47 interpolation operator, 148 inverse

Fourier transform, 33 image, 36 problem, 248

isometry, 69 partial 69

ket, 14 Kronecker delta, 5

least-energy representation, 101 least-squares approximation, 88 Lebesgue

integral, 21 measure, 35

light cone, 208 linear, 6

functional, 24 bounded 24

linearly dependent, 4, 80 independent, 4

Lipschitz continuous, 187 Lorentz

condition, 211 group, 259 invariant inner product, 208 transformations, 237

low-pass filter, 147

Index 299

massless Sobolev space, 216 matching, 248 Maxwell's equations, 207 mean time, 191 measurable, 21, 35 measure, 21, 35

space, 40 zero, 22

metric operator, 18 Mexican hat, 75 Meyer wavelets, 167 moments, 193 multiresolution analysis, 144

narrow-band signal, 252 negative-frequency cone, 222 nondiffracting beams, 272 norm, 12 numerically stable, 104 Nyquist rate, 101

operator, 23 bounded, 23

orthogonal, 13 orthonormal, 13 oscillation, 273 over complete, 4 oversampling, 88, 101

Parseval's identity, 33 past cone, 222 periodic, 26 Plancherel's theorem, 29, 32 plane waves, 206 Poincare group, 259 polarization identity, 33 positive-definite operators, 18 potentials, 211 proxies, 246, 256, 280

quadrature mirror filters, 148 quaternionic structure, 171

range, 245 ray filter, 243 redundancy, 108 reference wavelet, 239, 240n., 241 reflectivity distribution, 248

reflector complex, 267 elementary, 263 standard, 263

regular lattice, 117 representation, 206, 261 reproducing kernel, 70, 228-233 resolution, 145 resolution of unity, 10

continuous, 55 rest frame, 281-283 Riesz representation theorem, 24

sampling, 92, 99 rate, 101

sampling operator, 161 scale factor, 63 scaling

function, 140 invariant, 126 periodic, 127

Schwarz inequality, 18, 23 Shannon

sampling theorem, 99 wavelets, 167

space-time domain, 31 inversion, 260

span, 9 special conformal transformations, 212 spectral factorization, 180 square-integrable, 23 square-summable, 20 stability, 243

condition, 128 star notation, 17 subband, 111

filtering, 153-155 subspace, 4 support, 23

compact, 23 synthesizing operator, 85, 236

time-limited, 102 transform,

analytic-signal, 218 triangle inequality, 18, 23 trigonometric polynomial, 143

300 Index

tube, causal, 206, 223 future, 223 past, 223

uncertainty principle, 52 up-sampling operator, 148

vector space, 4 complex 4 real 4

wave equation, 207, 272 vector, 30

wavelet, 62 basic 62 mother 62 packets, 172 space, 236

wave number-frequency domain, 31 Weyl-Heisenberg group, 51, 205 window, 45

Zak transform, 117 zero moments, 178 zoom operators, 148