Wavelets and Other Orthogonal Systems

Wavelets and Other Orthogonal Systems

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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.show more

Product details

  • Hardback | 392 pages
  • 159 x 246.4 x 27.2mm | 734.83g
  • Taylor & Francis Inc
  • CRC Press Inc
  • Bosa Roca, United States
  • English
  • Revised
  • 2nd Revised edition
  • 25 black & white illustrations, 2 black & white tables
  • 1584882271
  • 9781584882275
  • 2,498,701

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 sectionshow more