| Introduction to wavelet analysis over R / Chapter 1： |
| A short motivation / 1： |
| Time-frequency analysis / 1.1： |
| Wavelets and approximation theory / 1.2： |
| Some easy properties of the wavelet transform / 2： |
| Wavelet transform in Fourier space / 3： |
| Co-variance of wavelet transforms / 4： |
| Voices, zooms, and convolutions / 5： |
| Laplace convolution / 5.1： |
| Scale convolution / 5.2： |
| Mellin transforms / 5.3： |
| The basic functions: the wavelets / 6： |
| The real wavelets / 7： |
| The progressive wavelets / 8： |
| Progressive wavelets with real-valued frequency representation / 8.1： |
| Chirp wavelets / 8.2： |
| On the modulus of progressive functions / 8.3： |
| Some explicit analysed functions and easy examples / 9： |
| The wavelet transform of pure frequencies / 9.1： |
| The real oscillations / 9.2： |
| The onsets / 9.4： |
| The wavelet analysis of a hyperbolic chirp / 9.5： |
| Interactions / 9.6： |
| Two deltas / 9.7： |
| Delta and pure frequency / 9.8： |
| The influence cone and easy localization properties / 10： |
| Polynomial localization / 11： |
| More precise results / 11.1： |
| The influence regions for pure frequencies / 12： |
| The space of highly time-frequency localized functions / 13： |
| The inversion formula / 14： |
| Fourier transform in wavelet space / 14.1： |
| Reconstruction with singular reconstruction wavelets / 15： |
| The wavelet synthesis operator / 16： |
| Reconstruction without reconstruction wavelet / 17： |
| Localization properties of the wavelet synthesis / 18： |
| Frequency localization / 18.1： |
| Time localization / 18.2： |
| Wavelet analysis over S[subscript +](R) / 19： |
| Schwartz space / 19.1： |
| The regularity of the image space / 19.2： |
| The reproducing kernel / 20： |
| The cross-kernel / 20.1： |
| The wavelet transform of a white noise / 21： |
| The wavelet transform in L[superscript 2](R) / 22： |
| The inverse wavelet transform / 23： |
| The wavelet transform over S[prime subscript +](R) / 24： |
| Definition of the wavelet transform / 24.1： |
| The wavelet transform on S[prime](R) / 25： |
| A class of operators / 26： |
| The derivation operator and Riesz potentials / 26.1： |
| Differentiation and integration over S[prime subscript 0](R) / 26.2： |
| Singular support of distributions / 27： |
| Bounded sets in S[subscript 0](R) and S[prime subscript 0](R) / 28： |
| Some explicit wavelet transforms of distributions / 29： |
| The distributions..., [Characters not reproducible] / 29.1： |
| The distributions [Characters not reproducible] / 29.2： |
| Extension to higher dimensions / 30： |
| Proof of Theorem 11.1.1 / 31： |
| Discretizing and periodizing the half-plane / Chapter 2： |
| Interpolation |
| Reconstruction over voices |
| One single voice / 2.1： |
| Infinitely many voices / 2.2： |
| An iteration procedure |
| Calderon-Zygmund operators: a first contact |
| Reconstruction over strips |
| The pointwise and uniform convergence of the inversion formula |
| Uniform convergence in L[superscript p](R), 1< p< [infinity] / 6.1： |
| Pointwise convergence in L[superscript p](R), 1 [greater than or equal] p< [infinity] / 6.2： |
| Pointwise convergence in L[superscript infinity](R) / 6.3： |
| The 'Gibbs' phenomenon for s[subscript epsilon, rho] |
| Gibbs phenomenon / 7.1： |
| No Gibbs phenomenon / 7.2： |
| Reconstruction over cones |
| The Poisson summation formula |
| Periodic functions |
| The periodizing operator |
| Sequences and sampling |
| The Fourier transform over the circle |
| Some sampling theorems |
| The continuous wavelet transform over T |
| Wavelet analysis of S(T) and S[prime](T) / 10.1： |
| The wavelet transform of L[superscript 2](T) / 10.2： |
| Sampling of voices |
| Frames and moments |
| Some wavelet frames |
| Irregular sampling / 13.1： |
| Calderon-Zygmund operators again / 13.2： |
| A functional calculus |
| The case of self-adjoint operators |
| The function e[superscript itA] / 14.2： |
| Multi-resolution analysis / Chapter 3： |
| Riesz bases |
| The Fourier space picture |
| Translation invariant orthonormal basis / 1.3： |
| Skew projections / 1.4： |
| Perfect sampling spaces / 1.5： |
| Splines / 1.6： |
| Exponential localization / 1.7： |
| Perfect sampling spaces of spline functions / 1.8： |
| Sampling spaces over Z, T, and Z/NZ |
| Sampling space over Z |
| Oversampling of sampling spaces / 2.3： |
| Sampling spaces over T / 2.4： |
| Periodizing a sampling space over R / 2.5： |
| Periodizing a sampling space over T / 2.6： |
| Sampling spaces over Z/NZ / 2.7： |
| Quadrature mirror filters in L[superscript 2](Z) |
| Completing a QMF-system / 3.1： |
| Complements over R / 3.2： |
| QMF over Z/NZ and complements over T / 3.3： |
| Multi-resolution analysis over R |
| Localization and regularity of [psi] / 4.1： |
| Examples of multi-resolution analysis and wavelets |
| The Haar system |
| Splines wavelets |
| Band-limited functions |
| Littlewood-Paley analysis / 5.4： |
| The partial reconstruction operator |
| Multi-resolution analysis of L[superscript 2](Z) |
| Isometrics and the shift operator |
| QMF and multi-resolution analysis over Z |
| Wavelets over Z / 7.3： |
| QMF and multi-resolution analysis |
| Compact support |
| An easy regularity estimate |
| The dyadic interpolation spaces |
| The Lagrange interpolation spaces |
| Compactly supported wavelets |
| Wavelet frames |
| Bi-orthogonal expansions |
| Bi-orthogonal expansions of L[superscript 2](Z) / 12.1： |
| Bi-orthogonal expansions in L[superscript 2](R) / 12.2： |
| QMF and loop groups |
| The group of unitary operators with [U, T[subscript 2]] = 0 |
| Some subclasses of QMF / 13.3： |
| The factorization problem / 13.5： |
| Multi-resolution analysis over T |
| Multi-resolution analysis over Z/2[superscript M]Z |
| Computing the discrete wavelet transform |
| Filterbanks over Z / 16.1： |
| Computing the orthonormal wavelet transform over a dyadic grid / 16.2： |
| More general wavelet / 16.3： |
| Denser grids / 16.4： |
| Interpolation of the voices / 16.5： |
| The 'a trous' algorithm / 16.6： |
| Computation over Z/2[superscript N]Z / 16.7： |
| Computing over R by using data over Z/NZ |
| Fractal analysis and wavelet transforms / Chapter 4： |
| Self-similarity and the re-normalization group |
| Re-normalization in wavelet-space |
| The order of magnitude of wavelet coefficients |
| Inverse theorems for global regularity |
| The class of Zygmund |
| Inverse theorems for local regularity |
| Pointwise differentiability and wavelet analysis |
| The class W[superscript alpha] |
| Asymptotic behaviour at small scales |
| The Brownian motion |
| The Weierstrass non-differentiable function |
| The Riemann-Weierstrass function |
| The orbit of 0 |
| The orbit of 1 |
| The non-degenerated fixed points |
| The irrational points / 6.4： |
| The baker's map |
| A family of dynamical systems and fractal measures |
| Self-similar fractal measure |
| The evolution in wavelet space |
| Some fractal measures |
| Fractal dimensions |
| Capacity |
| The generalized fractal dimensions |
| Fractal dimensions and wavelet transforms |
| Time evolution and the dimension [kappa](2) |
| Local self-similarity and singularities |
| The f([alpha]) spectrum |
| On the fractality of orthonormal wavelets |
| Group theory as unifying language / Chapter 5： |
| Some notions of group theory |
| Direct sum of groups |
| Quotient groups |
| Homomorphisms |
| Representations |
| Schur's lemma |
| Group action |
| Invariant measures |
| Regular representations |
| Group convolutions / 1.9： |
| Square integrable representations / 1.10： |
| The 'wavelet' analysis associated to square integrable representations |
| A priori estimates |
| Transformation properties |
| Energy conservation |
| The left- and right-synthesis |
| Co-variance |
| The inversion formulae |
| On the constant c[subscript g,h] |
| More general reconstruction |
| The reproducing kernel equation |
| Fourier transform over Abelian groups |
| The Fourier transform |
| Group-translations |
| The convolution theorem |
| Periodizing, sampling, and M. Poisson |
| Sampling |
| Periodization |
| Sampling spaces over Abelian groups |
| The discrete wavelet transform over Abelian groups |
| A group of operators |
| Polynomial loops: the factorization problem / 10.3： |
| The wavelet transform in two dimensions |
| Reconstruction formulae / 11.2： |
| A class of inverse problems / 11.3： |
| The Radon transform as wavelet transform |
| The Radon-inversion formula |
| Functional analysis and wavelets / Chapter 6： |
| Some function spaces |
| Wavelet multipliers |
| The class of highly regular Calderon-Zygmund operators (CZOs) |
| The dilation co-variance |
| Fourier multipliers as highly regular CZO |
| Singular integrals as highly regular CZO |
| Pointwise properties of highly regular CZO |
| Littlewood-Paley theory |
| The Sobolev spaces |
| Bibliography |
| Index |
| Introduction to wavelet analysis over R / Chapter 1： |
| A short motivation / 1： |
| Time-frequency analysis / 1.1： |
図書
