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 R | p. 1 |
A short motivation | p. 1 |
Time-frequency analysis | p. 1 |
Wavelets and approximation theory | p. 6 |
Some easy properties of the wavelet transform | p. 8 |
Wavelet transform in Fourier space | p. 9 |
Co-variance of wavelet transforms | p. 11 |
Voices, zooms, and convolutions | p. 14 |
Laplace convolution | p. 14 |
Scale convolution | p. 15 |
Mellin transforms | p. 16 |
The basic functions: the wavelets | p. 17 |
The real wavelets | p. 20 |
The progressive wavelets | p. 25 |
Progressive wavelets with real-valued frequency representation | p. 28 |
Chirp wavelets | p. 33 |
On the modulus of progressive functions | p. 35 |
Some explicit analysed functions and easy examples | p. 37 |
The wavelet transform of pure frequencies | p. 37 |
Chirp wavelets | p. 38 |
The real oscillations | p. 41 |
The onsets | p. 42 |
The wavelet analysis of a hyperbolic chirp | p. 45 |
Interactions | p. 49 |
Two deltas | p. 49 |
Delta and pure frequency | p. 53 |
The influence cone and easy localization properties | p. 53 |
Polynomial localization | p. 57 |
More precise results | p. 58 |
The influence regions for pure frequencies | p. 59 |
The space of highly time-frequency localized functions | p. 63 |
The inversion formula | p. 65 |
Fourier transform in wavelet space | p. 67 |
Reconstruction with singular reconstruction wavelets | p. 67 |
The wavelet synthesis operator | p. 68 |
Reconstruction without reconstruction wavelet | p. 70 |
Localization properties of the wavelet synthesis | p. 70 |
Frequency localization | p. 71 |
Time localization | p. 72 |
Wavelet analysis over S[subscript +](R) | p. 74 |
Schwartz space | p. 76 |
The regularity of the image space | p. 78 |
The reproducing kernel | p. 79 |
The cross-kernel | p. 81 |
The wavelet transform of a white noise | p. 83 |
The wavelet transform in L[superscript 2](R) | p. 84 |
The inverse wavelet transform | p. 89 |
The wavelet transform over S[prime subscript +](R) | p. 92 |
Definition of the wavelet transform | p. 96 |
The wavelet transform on S[prime](R) | p. 103 |
A class of operators | p. 104 |
The derivation operator and Riesz potentials | p. 105 |
Differentiation and integration over S[prime subscript 0](R) | p. 107 |
Singular support of distributions | p. 109 |
Bounded sets in S[subscript 0](R) and S[prime subscript 0](R) | p. 110 |
Some explicit wavelet transforms of distributions | p. 112 |
The distributions..., [Characters not reproducible] | p. 112 |
The distributions [Characters not reproducible] | p. 112 |
Extension to higher dimensions | p. 115 |
Proof of Theorem 11.1.1 | p. 117 |
Discretizing and periodizing the half-plane | p. 121 |
Interpolation | p. 121 |
Reconstruction over voices | p. 123 |
One single voice | p. 124 |
Infinitely many voices | p. 125 |
An iteration procedure | p. 128 |
Calderon-Zygmund operators: a first contact | p. 130 |
Reconstruction over strips | p. 131 |
The pointwise and uniform convergence of the inversion formula | p. 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 phenomenon | p. 144 |
No Gibbs phenomenon | p. 145 |
Reconstruction over cones | p. 145 |
The Poisson summation formula | p. 146 |
Periodic functions | p. 147 |
The periodizing operator | p. 148 |
Sequences and sampling | p. 149 |
The Fourier transform over the circle | p. 150 |
The Poisson summation formula | p. 151 |
Some sampling theorems | p. 152 |
The continuous wavelet transform over T | p. 154 |
Wavelet analysis of S(T) and S[prime](T) | p. 155 |
The wavelet transform of L[superscript 2](T) | p. 159 |
Sampling of voices | p. 160 |
Frames and moments | p. 161 |
Some wavelet frames | p. 166 |
Irregular sampling | p. 172 |
Calderon-Zygmund operators again | p. 175 |
A functional calculus | p. 176 |
The case of self-adjoint operators | p. 178 |
The function e[superscript itA] | p. 180 |
Multi-resolution analysis | p. 182 |
Some sampling theorems | p. 182 |
Riesz bases | p. 183 |
The Fourier space picture | p. 185 |
Translation invariant orthonormal basis | p. 187 |
Skew projections | p. 188 |
Perfect sampling spaces | p. 189 |
Splines | p. 192 |
Exponential localization | p. 193 |
Perfect sampling spaces of spline functions | p. 193 |
Sampling spaces over Z, T, and Z/NZ | p. 194 |
Sampling space over Z | p. 194 |
Skew projections | p. 197 |
Oversampling of sampling spaces | p. 197 |
Sampling spaces over T | p. 198 |
Periodizing a sampling space over R | p. 201 |
Periodizing a sampling space over T | p. 201 |
Sampling spaces over Z/NZ | p. 202 |
Quadrature mirror filters in L[superscript 2](Z) | p. 203 |
Completing a QMF-system | p. 205 |
Complements over R | p. 205 |
QMF over Z/NZ and complements over T | p. 206 |
Multi-resolution analysis over R | p. 206 |
Localization and regularity of [psi] | p. 208 |
Examples of multi-resolution analysis and wavelets | p. 209 |
The Haar system | p. 209 |
Splines wavelets | p. 209 |
Band-limited functions | p. 210 |
Littlewood-Paley analysis | p. 210 |
The partial reconstruction operator | p. 211 |
Multi-resolution analysis of L[superscript 2](Z) | p. 213 |
Isometrics and the shift operator | p. 217 |
QMF and multi-resolution analysis over Z | p. 218 |
Wavelets over Z | p. 220 |
QMF and multi-resolution analysis | p. 221 |
Compact support | p. 228 |
An easy regularity estimate | p. 228 |
The dyadic interpolation spaces | p. 230 |
The Lagrange interpolation spaces | p. 232 |
Compactly supported wavelets | p. 235 |
Wavelet frames | p. 238 |
Bi-orthogonal expansions | p. 239 |
Bi-orthogonal expansions of L[superscript 2](Z) | p. 240 |
Bi-orthogonal expansions in L[superscript 2](R) | p. 242 |
QMF and loop groups | p. 244 |
The group of unitary operators with [U, T[subscript 2]] = 0 | p. 244 |
The Fourier space picture | p. 245 |
QMF and loop groups | p. 247 |
Some subclasses of QMF | p. 248 |
The factorization problem | p. 249 |
Multi-resolution analysis over T | p. 252 |
Multi-resolution analysis over Z/2[superscript M]Z | p. 255 |
Computing the discrete wavelet transform | p. 259 |
Filterbanks over Z | p. 261 |
Computing the orthonormal wavelet transform over a dyadic grid | p. 262 |
More general wavelet | p. 263 |
Denser grids | p. 264 |
Interpolation of the voices | p. 265 |
The 'a trous' algorithm | p. 266 |
Computation over Z/2[superscript N]Z | p. 268 |
Computing over R by using data over Z/NZ | p. 269 |
Fractal analysis and wavelet transforms | p. 271 |
Self-similarity and the re-normalization group | p. 271 |
Re-normalization in wavelet-space | p. 274 |
The order of magnitude of wavelet coefficients | p. 275 |
Inverse theorems for global regularity | p. 279 |
The class of Zygmund | p. 284 |
Inverse theorems for local regularity | p. 286 |
Pointwise differentiability and wavelet analysis | p. 289 |
The class W[superscript alpha] | p. 292 |
Asymptotic behaviour at small scales | p. 293 |
The Brownian motion | p. 296 |
The Weierstrass non-differentiable function | p. 299 |
The Riemann-Weierstrass function | p. 300 |
The orbit of 0 | p. 307 |
The orbit of 1 | p. 309 |
The non-degenerated fixed points | p. 312 |
The irrational points | p. 313 |
The baker's map | p. 314 |
A family of dynamical systems and fractal measures | p. 316 |
Self-similar fractal measure | p. 319 |
The evolution in wavelet space | p. 320 |
Some fractal measures | p. 321 |
Fractal dimensions | p. 324 |
Capacity | p. 324 |
The generalized fractal dimensions | p. 329 |
Fractal dimensions and wavelet transforms | p. 331 |
Time evolution and the dimension [kappa](2) | p. 336 |
Local self-similarity and singularities | p. 339 |
The f([alpha]) spectrum | p. 340 |
On the fractality of orthonormal wavelets | p. 341 |
Group theory as unifying language | p. 344 |
Some notions of group theory | p. 344 |
Direct sum of groups | p. 345 |
Quotient groups | p. 346 |
Homomorphisms | p. 347 |
Representations | p. 348 |
Schur's lemma | p. 351 |
Group action | p. 353 |
Invariant measures | p. 353 |
Regular representations | p. 355 |
Group convolutions | p. 357 |
Square integrable representations | p. 357 |
The 'wavelet' analysis associated to square integrable representations | p. 359 |
A priori estimates | p. 360 |
Transformation properties | p. 360 |
Energy conservation | p. 361 |
The left- and right-synthesis | p. 363 |
Co-variance | p. 363 |
The inversion formulae | p. 364 |
On the constant c[subscript g,h] | p. 365 |
More general reconstruction | p. 366 |
The reproducing kernel equation | p. 368 |
Fourier transform over Abelian groups | p. 369 |
The Fourier transform | p. 371 |
Group-translations | p. 372 |
The convolution theorem | p. 372 |
Periodizing, sampling, and M. Poisson | p. 373 |
Sampling | p. 373 |
Periodization | p. 374 |
The Poisson summation formula | p. 374 |
Sampling spaces over Abelian groups | p. 376 |
The discrete wavelet transform over Abelian groups | p. 378 |
A group of operators | p. 379 |
The Fourier space picture | p. 380 |
QMF and loop groups | p. 381 |
Polynomial loops: the factorization problem | p. 384 |
The wavelet transform in two dimensions | p. 384 |
Energy conservation | p. 387 |
Reconstruction formulae | p. 387 |
The inversion formula | p. 388 |
A class of inverse problems | p. 391 |
The Radon transform as wavelet transform | p. 392 |
The Radon-inversion formula | p. 393 |
Functional analysis and wavelets | p. 396 |
Some function spaces | p. 396 |
Wavelet multipliers | p. 401 |
The class of highly regular Calderon-Zygmund operators (CZOs) | p. 402 |
The dilation co-variance | p. 404 |
Fourier multipliers as highly regular CZO | p. 405 |
Singular integrals as highly regular CZO | p. 406 |
Pointwise properties of highly regular CZO | p. 409 |
Littlewood-Paley theory | p. 410 |
The Sobolev spaces | p. 412 |
Bibliography | p. 415 |
Index | p. 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