Wavelets and Other Orthogonal Systems: 2nd Edition (Hardback) book cover

Wavelets and Other Orthogonal Systems

2nd Edition

By Gilbert G. Walter, Xiaoping Shen

CRC Press

392 pages | 25 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781584882275
pub: 2000-12-20
SAVE ~$21.75
$145.00
$123.25
x
eBook (VitalSource) : 9781315273716
pub: 2018-10-03
from $28.98


FREE Standard Shipping!

Description

A bestseller in its first edition, Wavelets and Other Orthogonal Systems: Second Edition has been fully updated to reflect the recent growth and development of this field, especially in the area of multiwavelets. The authors have incorporated more examples and numerous illustrations to help clarify concepts. They have also added a considerable amount of new material, including sections addressing impulse trains, an alternate approach to periodic wavelets, and positive wavelet s. Other new discussions include irregular sampling in wavelet subspaces, hybrid wavelet sampling, interpolating multiwavelets, and several new statistics topics.

With cutting-edge applications in data compression, image analysis, numerical analysis, and acoustics wavelets remain at the forefront of current research. Wavelets and Other Orthogonal Systems maintains its mathematical perspective in presenting wavelets in the same setting as other orthogonal systems, thus allowing their advantages and disadvantages to be seen more directly. Now even more student friendly, the second edition forms an outstanding text not only for graduate students in mathematics, but also for those interested in scientific and engineering applications.

Table of Contents

ORTHOGONAL SERIES

General Theory

Examples

A PRIMER ON TEMPERED DISTRIBUTIONS

Intuitive Introduction

Test Functions

Tempered Distribution

Fourier Transforms

Periodic Distributions

Analytic Representations

Sobolev Spaces

AN INTRODUCTION TO ORTHOGONAL WAVELET THEORY

Multiresolution Analysis

Mother Wavelet

Reproducing Kernels and a Moment Condition

Regularity of Wavelets as a Moment Condition

Mallat's Decomposition and Reconstruction Algorithm

Filters

CONVERGENCE AND SUMMABILITY OF FOURIER SERIES

Pointwise Convergence

Summability

Gibbs' Phenomenon

Periodic Distributions

WAVELETS AND TEMPERED DISTRIBUTIONS

Multiresolution Analysis of Tempered Distributions

Wavelets Based on Distributions

Distributions with Point Support

Approximation with Impulse Trains

ORTHOGONAL POLYNOMIALS

General Theory

Classical Orthogonal Polynomials

Problems

OTHER ORTHOGONAL SYSTEMS

Self-Adjoint Eigenvalue Problems on Finite Intervals

Hilber-Schmnidt Integral Operators

An Anomaly: The Prolate Spheroidal Functions

A Lucky Accident?

Rademacher Functions

Walsh Function

Periodic Wavelets

Local Sine or Cosine Base

Biorthogonal Wavelets

POINTWISE CONVERGENCE OF WAVELET EXPANSIONS

Reproducing Kernel Delta Sequences

Positive and Quasi-Positive Delta Sequences

Local Convergence of Distribution Expansions

Convergence Almost Everywhere

Rate of Convergence of the Delta Sequence

Other Partial Sums of the Wavelet Expansion

Gibbs' Phenomenon

Positive Scaling Functions

A SHANNON SAMPLING THEOREM IN WAVELET SUBSPACES

A Riesz Basis of Vm

The Ampling Sequence in Vm

Examples of Sampling theorems

The Sampling Sequence in Tm

Shifted Sampling

Gibbs' Phenomenon for Sampling Series

Irregular Sampling in Wavelet Subspaces

EXTENSIONS OF WAVELET SAMPLING THEOREMS

Oversampling with Scaling Functions

Hybrid Sampling Series

Positive Hybrid Sampling

The Convergence of the Positive Hybrid Series

Cardinal Scaling Functions

Interpolating Multiwavelets

Orthogonal Finite Element Multiwavelets

TRANSLATION AND DILATION INVARIANCE IN ORTHOGONAL SYSTEMS

Trigonometric System

Orthogonal Polynomials

An Example Where Everything Works

An Example Where Nothing Works

Weak Translation Invariance

Dilations and Other Operations

ANALYTIC REPRESENTATIONS VIA ORTHOGONAL SERIES

Trigonometric Series

Hermite Series

Legendre Polynomial Series

Analytic and Harmonic Wavelets

Analytic Solutions to Dilation Equations

Analytic Representation of Distributions by Wavelets

Wavelets Analytic in the Entire Complex Plane

ORTHOGONAL SERIES IN STATISTICS

Fourier Series Density Estimators

Hermite Series Density Estimators

The Histogram as a Wavelet Estimator

Smooth Wavelet Estimators of Density

Local Convergence

Positive Density Estimators Based on Characteristic Functions

Positive Estimators Based on Positive Wavelets

Density Estimation with Noisy Data

Other Estimation with Wavelets

Threshold Methods

ORTHOGONAL SYSTEMS AND STOCHASTIC PROCESSES

K-L Expansions

Stationary Processes and Wavelets

A Series with Uncorrelated Coefficients

Wavelets Based on Band Limited Processes

Nonstationary Processes

Each chapter also contains a Problems section

About the Series

Studies in Advanced Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations