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Wavelets : An Analysis Tool - Matthias Holschneider

Wavelets

An Analysis Tool

Hardcover Published: 1st October 1998
ISBN: 9780198505211
Number Of Pages: 423

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Wavelets analysis--a new and rapidly growing field of research--has been applied to a wide range of endeavors, from signal data analysis (geoprospection, speech recognition, and singularity detection) to data compression (image and voice-signals) to pure mathematics. Written in an accessible, user-friendly style, Wavelets: An Analysis Tool offers a self-contained, example-packed introduction to the subject. Taking into account the continuous transform as well as its discretized version (the ortho-normal basis) the book begins by introducing the continuous wavelets transform in one dimension. It goes on to provide detailed discussions of wavelet analysis of regular functions, tempered distributions, square integrable functions, and the continuous wavelet transform. Throughout, the language of group theory is used to unify various approaches. Profusely illustrated and containing information not available elsewhere, this book is ideal for advanced students and researchers in mathematics, physics, and signal processing engineering.

"The book could serve well as a text, reference or tutorial on wavelets. It has a strong theoretical component and contains the foundations for the most current applications of wavelet theory." --Physics Today"A nicely written, fairly self-contained introduction. . .Statisticians might be interested in the very well written introduction (over 100 pages) and the parts of the book that concentrate on the wavelet as a time-frequency analysis tool."--Journal of the American Statistical Association "The book could serve well as a text, reference or tutorial on wavelets. It has a strong theoretical component and contains the foundations for the most current applications of wavelet theory." --Physics Today "A nicely written, fairly self-contained introduction. . .Statisticians might be interested in the very well written introduction (over 100 pages) and the parts of the book that concentrate on the wavelet as a time-frequency analysis tool."--Journal of the American Statistical Association "The book could serve well as a text, reference or tutorial on wavelets. It has a strong theoretical component and contains the foundations for the most current applications of wavelet theory." --Physics Today "A nicely written, fairly self-contained introduction. . .Statisticians might be interested in the very well written introduction (over 100 pages) and the parts of the book that concentrate on the wavelet as a time-frequency analysis tool."--Journal of the American Statistical Association "The book could serve well as a text, reference or tutorial on wavelets. It has a strong theoretical component and contains the foundations for the most current applications of wavelet theory." --Physics Today"A nicely written, fairly self-contained introduction. . .Statisticians might be interested in the very well written introduction (over 100 pages) and the parts of the book that concentrate on the wavelet as a time-frequency analysis tool."--Journal of the American Statistical Association

Introduction to wavelet analysis over Rp. 1
A short motivationp. 1
Time-frequency analysisp. 1
Wavelets and approximation theoryp. 6
Some easy properties of the wavelet transformp. 8
Wavelet transform in Fourier spacep. 9
Co-variance of wavelet transformsp. 11
Voices, zooms, and convolutionsp. 14
Laplace convolutionp. 14
Scale convolutionp. 15
Mellin transformsp. 16
The basic functions: the waveletsp. 17
The real waveletsp. 20
The progressive waveletsp. 25
Progressive wavelets with real-valued frequency representationp. 28
Chirp waveletsp. 33
On the modulus of progressive functionsp. 35
Some explicit analysed functions and easy examplesp. 37
The wavelet transform of pure frequenciesp. 37
Chirp waveletsp. 38
The real oscillationsp. 41
The onsetsp. 42
The wavelet analysis of a hyperbolic chirpp. 45
Interactionsp. 49
Two deltasp. 49
Delta and pure frequencyp. 53
The influence cone and easy localization propertiesp. 53
Polynomial localizationp. 57
More precise resultsp. 58
The influence regions for pure frequenciesp. 59
The space of highly time-frequency localized functionsp. 63
The inversion formulap. 65
Fourier transform in wavelet spacep. 67
Reconstruction with singular reconstruction waveletsp. 67
The wavelet synthesis operatorp. 68
Reconstruction without reconstruction waveletp. 70
Localization properties of the wavelet synthesisp. 70
Frequency localizationp. 71
Time localizationp. 72
Wavelet analysis over S[subscript +](R)p. 74
Schwartz spacep. 76
The regularity of the image spacep. 78
The reproducing kernelp. 79
The cross-kernelp. 81
The wavelet transform of a white noisep. 83
The wavelet transform in L[superscript 2](R)p. 84
The inverse wavelet transformp. 89
The wavelet transform over S[prime subscript +](R)p. 92
Definition of the wavelet transformp. 96
The wavelet transform on S[prime](R)p. 103
A class of operatorsp. 104
The derivation operator and Riesz potentialsp. 105
Differentiation and integration over S[prime subscript 0](R)p. 107
Singular support of distributionsp. 109
Bounded sets in S[subscript 0](R) and S[prime subscript 0](R)p. 110
Some explicit wavelet transforms of distributionsp. 112
The distributions..., [Characters not reproducible]p. 112
The distributions [Characters not reproducible]p. 112
Extension to higher dimensionsp. 115
Proof of Theorem 11.1.1p. 117
Discretizing and periodizing the half-planep. 121
Interpolationp. 121
Reconstruction over voicesp. 123
One single voicep. 124
Infinitely many voicesp. 125
An iteration procedurep. 128
Calderon-Zygmund operators: a first contactp. 130
Reconstruction over stripsp. 131
The pointwise and uniform convergence of the inversion formulap. 134
Uniform convergence in L[superscript p](R), 1< p< [infinity]p. 136
Pointwise convergence in L[superscript p](R), 1 [greater than or equal] p< [infinity]p. 140
Pointwise convergence in L[superscript infinity](R)p. 141
The 'Gibbs' phenomenon for s[subscript epsilon, rho]p. 142
Gibbs phenomenonp. 144
No Gibbs phenomenonp. 145
Reconstruction over conesp. 145
The Poisson summation formulap. 146
Periodic functionsp. 147
The periodizing operatorp. 148
Sequences and samplingp. 149
The Fourier transform over the circlep. 150
The Poisson summation formulap. 151
Some sampling theoremsp. 152
The continuous wavelet transform over Tp. 154
Wavelet analysis of S(T) and S[prime](T)p. 155
The wavelet transform of L[superscript 2](T)p. 159
Sampling of voicesp. 160
Frames and momentsp. 161
Some wavelet framesp. 166
Irregular samplingp. 172
Calderon-Zygmund operators againp. 175
A functional calculusp. 176
The case of self-adjoint operatorsp. 178
The function e[superscript itA]p. 180
Multi-resolution analysisp. 182
Some sampling theoremsp. 182
Riesz basesp. 183
The Fourier space picturep. 185
Translation invariant orthonormal basisp. 187
Skew projectionsp. 188
Perfect sampling spacesp. 189
Splinesp. 192
Exponential localizationp. 193
Perfect sampling spaces of spline functionsp. 193
Sampling spaces over Z, T, and Z/NZp. 194
Sampling space over Zp. 194
Skew projectionsp. 197
Oversampling of sampling spacesp. 197
Sampling spaces over Tp. 198
Periodizing a sampling space over Rp. 201
Periodizing a sampling space over Tp. 201
Sampling spaces over Z/NZp. 202
Quadrature mirror filters in L[superscript 2](Z)p. 203
Completing a QMF-systemp. 205
Complements over Rp. 205
QMF over Z/NZ and complements over Tp. 206
Multi-resolution analysis over Rp. 206
Localization and regularity of [psi]p. 208
Examples of multi-resolution analysis and waveletsp. 209
The Haar systemp. 209
Splines waveletsp. 209
Band-limited functionsp. 210
Littlewood-Paley analysisp. 210
The partial reconstruction operatorp. 211
Multi-resolution analysis of L[superscript 2](Z)p. 213
Isometrics and the shift operatorp. 217
QMF and multi-resolution analysis over Zp. 218
Wavelets over Zp. 220
QMF and multi-resolution analysisp. 221
Compact supportp. 228
An easy regularity estimatep. 228
The dyadic interpolation spacesp. 230
The Lagrange interpolation spacesp. 232
Compactly supported waveletsp. 235
Wavelet framesp. 238
Bi-orthogonal expansionsp. 239
Bi-orthogonal expansions of L[superscript 2](Z)p. 240
Bi-orthogonal expansions in L[superscript 2](R)p. 242
QMF and loop groupsp. 244
The group of unitary operators with [U, T[subscript 2]] = 0p. 244
The Fourier space picturep. 245
QMF and loop groupsp. 247
Some subclasses of QMFp. 248
The factorization problemp. 249
Multi-resolution analysis over Tp. 252
Multi-resolution analysis over Z/2[superscript M]Zp. 255
Computing the discrete wavelet transformp. 259
Filterbanks over Zp. 261
Computing the orthonormal wavelet transform over a dyadic gridp. 262
More general waveletp. 263
Denser gridsp. 264
Interpolation of the voicesp. 265
The 'a trous' algorithmp. 266
Computation over Z/2[superscript N]Zp. 268
Computing over R by using data over Z/NZp. 269
Fractal analysis and wavelet transformsp. 271
Self-similarity and the re-normalization groupp. 271
Re-normalization in wavelet-spacep. 274
The order of magnitude of wavelet coefficientsp. 275
Inverse theorems for global regularityp. 279
The class of Zygmundp. 284
Inverse theorems for local regularityp. 286
Pointwise differentiability and wavelet analysisp. 289
The class W[superscript alpha]p. 292
Asymptotic behaviour at small scalesp. 293
The Brownian motionp. 296
The Weierstrass non-differentiable functionp. 299
The Riemann-Weierstrass functionp. 300
The orbit of 0p. 307
The orbit of 1p. 309
The non-degenerated fixed pointsp. 312
The irrational pointsp. 313
The baker's mapp. 314
A family of dynamical systems and fractal measuresp. 316
Self-similar fractal measurep. 319
The evolution in wavelet spacep. 320
Some fractal measuresp. 321
Fractal dimensionsp. 324
Capacityp. 324
The generalized fractal dimensionsp. 329
Fractal dimensions and wavelet transformsp. 331
Time evolution and the dimension [kappa](2)p. 336
Local self-similarity and singularitiesp. 339
The f([alpha]) spectrump. 340
On the fractality of orthonormal waveletsp. 341
Group theory as unifying languagep. 344
Some notions of group theoryp. 344
Direct sum of groupsp. 345
Quotient groupsp. 346
Homomorphismsp. 347
Representationsp. 348
Schur's lemmap. 351
Group actionp. 353
Invariant measuresp. 353
Regular representationsp. 355
Group convolutionsp. 357
Square integrable representationsp. 357
The 'wavelet' analysis associated to square integrable representationsp. 359
A priori estimatesp. 360
Transformation propertiesp. 360
Energy conservationp. 361
The left- and right-synthesisp. 363
Co-variancep. 363
The inversion formulaep. 364
On the constant c[subscript g,h]p. 365
More general reconstructionp. 366
The reproducing kernel equationp. 368
Fourier transform over Abelian groupsp. 369
The Fourier transformp. 371
Group-translationsp. 372
The convolution theoremp. 372
Periodizing, sampling, and M. Poissonp. 373
Samplingp. 373
Periodizationp. 374
The Poisson summation formulap. 374
Sampling spaces over Abelian groupsp. 376
The discrete wavelet transform over Abelian groupsp. 378
A group of operatorsp. 379
The Fourier space picturep. 380
QMF and loop groupsp. 381
Polynomial loops: the factorization problemp. 384
The wavelet transform in two dimensionsp. 384
Energy conservationp. 387
Reconstruction formulaep. 387
The inversion formulap. 388
A class of inverse problemsp. 391
The Radon transform as wavelet transformp. 392
The Radon-inversion formulap. 393
Functional analysis and waveletsp. 396
Some function spacesp. 396
Wavelet multipliersp. 401
The class of highly regular Calderon-Zygmund operators (CZOs)p. 402
The dilation co-variancep. 404
Fourier multipliers as highly regular CZOp. 405
Singular integrals as highly regular CZOp. 406
Pointwise properties of highly regular CZOp. 409
Littlewood-Paley theoryp. 410
The Sobolev spacesp. 412
Bibliographyp. 415
Indexp. 421
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780198505211
ISBN-10: 0198505213
Series: Oxford Mathematical Monographs
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 423
Published: 1st October 1998
Country of Publication: GB
Dimensions (cm): 23.37 x 16.13  x 2.52
Weight (kg): 0.69