Wavelets and other orthogonal systems with applications

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Bibliographic Information

Wavelets and other orthogonal systems with applications

Gilbert G. Walter

CRC Press, c1994

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Includes bibliographical references and index

Description and Table of Contents

Description

This book makes accessible to both mathematicians and engineers important elements of the theory, construction, and application of orthogonal wavelets. It is integrated with more traditional orthogonal series, such as Fourier series and orthogonal polynomials. It treats the interaction of both with generalized functions (delta functions), which have played an important part in engineering theory but whose rules are often vaguely presented. Unlike most other books that are excessively technical, this text/reference presents the basic concepts and examples in a readable form. Much of the material on wavelets has not appeared previously in book form. Applications to statistics, sampling theorems, and stochastic processes are given. In particular, the close affinity between wavelets and sampling theorems is explained and developed.

Table of Contents

Chapter 1. Orthogonal Series General Theory Examples Problems Chapter 2. A Primer on Tempered Distributions Tempered Distributions Fourier Transforms Periodic Distributions Analytic Representations Sobolev Spaces Problems Chapter 3. 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 Problems Chapter 4. Convergence and Summability of Fourier Series Pointwise Convergence Summability Gibbs' Phenomenon Periodic Distributions Problems Chapter 5. Wavelets and Tempered Distributions Multiresolution Analysis of Tempered Distributions Wavelets Based on Distributions Distributions with Point Support Problems Chapter 6. Orthogonal Polynomials General Theory Classical Orthogonal Polynomials Problems Chapter 7. Other Orthogonal Systems Self Adjoint Eigenvalue Problems on a Finite Interval Hilbert-Schmidt Integral Operators An Anomaly-The Prolate Spheroidal Functions A Lucky Accident? Rademacher Functions Walsh Functions Periodic Wavelets Local Sine or Cosine Bases Biorthogonal Wavelets Problems Chapter 8. Pointwise Convergence of Wavelet Expansions 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 Problems Chapter 9. A Shannon Sampling Theorem in Vm A Riesz Basis of Vm The Sampling Sequence in Vm Examples of Sampling Theorems The Sampling Sequence in Tm Shifted Sampling Oversampling with Scaling Functions Cardinal Scaling Functions Problems Chapter 10. 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 Problems Chapter 11. Analytic Representations via Orthogonal Series Trigonometric Series Hermite Series Legendre Polynomial Series Analytical and Harmonic Wavelets Analytic Solutions to Dilation Equations Analytic Representation of Distributions by Wavelets Problems Chapter 12. 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 Other Estimation with Wavelets Problems Chapter 13. 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 Problems Bibliography Index

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