Wavelets and Other Orthogonal Systems with Applications

Buch | Hardcover

272 Seiten

1994
Crc Press Inc (Verlag)
978-0-8493-7878-2 (ISBN)

Infos

Discusses the important elements of the theory, construction and applications of orthogonal wavelets. These include applications to statistics, sampling theorems and stochastic processes. In particular, the close affinity between wavelets and sampling theorems is explained and developed.

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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.

Autor

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

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 13.7.1994
• Reihe/Serie: Studies in Advanced Mathematics
• Zusatzinfo: 1 Tables, black and white
• Verlagsort: Bosa Roca
• Sprache: englisch
• Maße: 156 x 235 mm
• Gewicht: 544 g
• Einbandart: gebunden
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Mechanik
• ISBN-10: 0-8493-7878-8 / 0849378788
• ISBN-13: 978-0-8493-7878-2 / 9780849378782
• Zustand: Neuware