瓦尔纳编著的《小波分析导论（英文影印版）》是一部全面讲述小波分析理论的基础教程，主要内容包括小波基的构造和分析。通过详细讲述哈尔级数展开了小波理论中心思想的讨论，进而运用更加抽象的方法讲述哈尔级数，更深层次的讲述了哈尔结构的变化和扩展。目次：（第一部分）基础：函数和收敛；傅里叶级数；傅里叶变换；信号和系统；（第二部分）哈尔系统：离散哈尔变换；（第三部分）正交小波基：光滑、紧支撑小波包；（第四部分）其他小波结构：双正交小波；小波包；（第五部分）应用：图像压缩；积分算子。附录。

preface

Ⅰ preliminaries

 1 functions and convergence

1.1 functions

 1.1.1 bounded (l∞) functions

 1.1.2 integrable (l1) functions

 1.1.3 square integrable (l2) functions

 1.1.4 differentiable (cn) functions

1.2 convergence of sequences of functions

 1.2.1 numerical convergence

 1.2.2 pointwise convergence

 1.2.3 uniform (l∞) convergence

 1.2.4 mean (ll) convergence

 1.2.5 mean-square (l2) convergence

 1.2.6 interchange of limits and integrals

 2 fourier series

2.1 trigonometric series

 2.1.1 periodic functions

 2.1.2 the trigonometric system

 2.1.3 the fourier coefficients

 2.1.4 convergence of fourier series

2.2 approximate identities

 2.2.1 motivation from fourier series

 2.2.2 definition and examples

 2.2.3 convergence theorems

2.3 generalized fourier series

 2.3.1 orthogonality

 2.3.2 generalized fourier series

 2.3.3 completeness

 3 the fourier transform

3.1 motivation and definition

3.2 basic properties of the fourier transform

3.3 fourier inversion

3.4 convolution

3.5 plancherel's formula

3.6 the fourier transform for l2 functions

3.7 smoothness versus decay

3.8 dilation, translation, and modulation

3.9 bandlimited functions and the sampling formula

 4 signals and systems

4.1 signals

4.2 systems

 4.2.1 causality and stability

4.3 periodic signals and the discrete fourier transform

 4.3.1 the discrete fourier transform

4.4 the fast fourier transform

4.5 l2 fourier series

Ⅱ the haar system

  5 the haar system

5.1 dyadic step functions

 5.1.1 the dyadic intervals

 5.1.2 the scale j dyadic step functions

5.2 the haar system

 5.2.1 the haar scaling functions and the haar functions.

 5.2.2 orthogonality of the haar system

 5.2.3 the splitting lemma

5.3 haar bases on [0, 1]

5.4 comparison of haar series with fourier series

 5.4.1 representation of functions with small support

 5.4.2 behavior of haar coefficients near jump discontinuities

 5.4.3 haar coefficients and global smoothness

5.5 haar bases on r

 5.5.1 the approximation and detail operators

 5.5.2 the scale j haar system on r

 5.5.3 the hair system on r

 6 the discrete haar transform

6.1 motivation

 6.1.1 the discrete haar transform (dht)

6.2 the dht in two dimensions

 6.2.1 the row-wise and column-wise approximations and details

 6.2.2 the dht for matrices

6.3 image analysis with the dht

 6.3.1 approximation and blurring

 6.3.2 horizontal, vertical, and diagonal edges

 6.3.3 "naive" image compression

Ⅲ orthonormal wavelet bases

 7 multiresolution analysis

7.1 orthonormal systems of translates

7.2 definition of multiresolution analysis

 7.2.1 some basic properties of mras

7.3 examples of multiresolution analysis

 7.3.1 the haar mra

 7.3.2 the piecewise linear mra

 7.3.3 the bandlimited mra

 7.3.4 the meyer mra

7.4 construction and examples of orthonormal wavelet bases

 7.4.1 examples of wavelet bases

 7.4.2 wavelets in two dimensions

 7.4.3 localization of wavelet bases

7.5 proof of theorem 7.35

 7.5.1 sufficient conditions for a wavelet basis

 7.5.2 proof of theorem 7.35

7.6 necessary properties of the scaling function

7.7 general spline wavelets

 7.7.1 basic properties of spline functions

 7.7.2 spline multiresolution analyses

 8 the discrete wavelet transform

8.1 motivation: from mra to a discrete transform

8.2 the quadrature mirror filter conditions

 8.2.1 motivation from mra

 8.2.2 the approximation and detail operators and their adjoints

 8.2.3 the quadrature mirror filter (qmf) conditions

8.3 the discrete wavelet transform (dwt)

 8.3.1 the dwt for signals

 8.3.2 the dwt for finite signals

 8.3.3 the dwt as an orthogonal transformation

8.4 scaling functions from scaling sequences

 8.4.1 the infinite product formula

 8.4.2 the cascade algorithm

 8.4.3 the support of the scaling function

 9 smooth, compactly supported wavelets

9.1 vanishing moments

 9.1.1 vanishing moments and smoothness

 9.1.2 vanishing moments and approximation

 9.1.3 vanishing moments and the reproduction of polynomials

 9.1.4 equivalent conditions for vanishing moments

9.2 the daubechies wavelets

 9.2.1 the daubechies polynomials

 9.2.2 spectral factorization

9.3 image analysis with smooth wavelets

 9.3.1 approximation and blurring

 9.3.2 "naive" image compression with smooth wavelets

Ⅳ other wavelet constructions

 10 biorthogonal wavelets

10.1 linear independence and biorthogonality

10.2 riesz bases and the frame condition

10.3 riesz bases of translates

10.4 generalized multiresolution analysis (gmra)

 10.4.1 basic properties of gmra

 10.4.2 dual gmra and riesz bases of wavelets

10.5 riesz bases orthogonal across scales

 10.5.1 example: the piecewise linear gmra

10.6 a discrete transform for biorthogonal wavelets

 10.6.1 motivation from gmra

 10.6.2 the qmf conditions

10.7 compactly supported biorthogonal wavelets

 10.7.1 compactly supported spline wavelets

 10.7.2 symmetric biorthogonal wavelets

 10.7.3 using symmetry in the dwt

 11 wavelet packets

11.1 motivation: completing the wavelet tree

11.2 localization of wavelet packets

 11.2.1 time/spatial localization

 11.2.2 frequency localization

11.3 0rthogonality and completeness properties of wavelet packets

 11.3.1 wavelet packet bases with a fixed scale

 11.3.2 wavelet packets with mixed scales

11.4 the discrete wavelet packet transform (dwpt)

 11.4.1 the dwpt for signals

 11.4.2 the dwpt for finite signals

11.5 the best-basis algorithm

 11.5.1 the discrete wavelet packet library

 11.5.2 the idea of the best basis

 11.5.3 description of the algorithm

Ⅴ applications

 12 image compression

12.1 the transform step

 12.1.1 wavelets or wavelet packets?

 12.1.2 choosing a filter

12.2 the quantization step

12.3 the coding step

 12.3.1 sources and codes

 12.3.2 entropy and information

 12.3.3 coding and compression

12.4 the binary huffman code

12.5 a model wavelet transform image coder

 12.5.1 examples

 13 integral operators

13.1 examples of integral operators

 13.1.1 sturm-liouville boundary value problems

 13.1.2 the hilbert transform

 13.1.3 the radon transform

13.2 the bcr algorithm

 13.2.1 the scale j approximation to t

 13.2.2 description of the algorithm

Ⅵ appendixes

 a review of advanced calculus and linear algebra

a.1 glossary of basic terms from advanced calculus and linear algebra

a.2 basic theorems from advanced calculus

 b excursions in wavelet theory

b.1 other wavelet constructions

 b.1.1 m-band wavelets

 b.1.2 wavelets with rational noninteger dilation factors

 b.1.3 local cosine bases

 b.1.4 the continuous wavelet transform

 b.1.5 non~mra wavelets

 b.1.6 multiwavelets

b.2 wavelets in other domains

 b.2.1 wavelets on intervals

 b.2.2' wavelets in higher dimensions

 b.2.3 the lifting scheme

b.3 applications of wavelets

 b.3.1 wavelet denoising

 b.3.2 multiscale edge detection

 b.3.3 the fbi fingerprint compression standard

 c references cited in the text

index