Wavelets and Other Orthogonal Systems with Applications

Walter; Gilbert G. University of Wisconsin, Milwaukee, Wisconsin, USA,

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