Approximation Theory From Taylor Polynomials to Wavelets

Approximation Theory From Taylor Polynomials to Wavelets / by Ole Christensen, Khadija Laghrida Christensen.

This concisely written book gives an elementary introduction to a classical area of mathematics—approximation theory—in a way that naturally leads to the modern field of wavelets. The exposition, driven by ideas rather than technical details and proofs, demonstrates the dynamic nature of mathematics...

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Bibliographic Details
Main Authors: Christensen, Ole (Author), Christensen, Khadija Laghrida (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2005.
Edition:1st ed. 2005.
Series:Applied and Numerical Harmonic Analysis,
Springer eBook Collection.
Subjects:
Fourier analysis.
Approximation theory.
Harmonic analysis.
Functional analysis.
Applied mathematics.
Engineering mathematics.
Signal processing.
Image processing.
Speech processing systems.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 1 Approximation with Polynomials
  • 1.1 Approximation of a function on an interval
  • 1.2 Weierstrass’ theorem
  • 1.3 Taylor’s theorem
  • 1.4 Exercises
  • 2 Infinite Series
  • 2.1 Infinite series of numbers
  • 2.2 Estimating the sum of an infinite series
  • 2.3 Geometric series
  • 2.4 Power series
  • 2.5 General infinite sums of functions
  • 2.6 Uniform convergence
  • 2.7 Signal transmission
  • 2.8 Exercises
  • 3 Fourier Analysis
  • 3.1 Fourier series
  • 3.2 Fourier’s theorem and approximation
  • 3.3 Fourier series and signal analysis
  • 3.4 Fourier series and Hilbert spaces
  • 3.5 Fourier series in complex form
  • 3.6 Parseval’s theorem
  • 3.7 Regularity and decay of the Fourier coefficients
  • 3.8 Best N-term approximation
  • 3.9 The Fourier transform
  • 3.10 Exercises
  • 4 Wavelets and Applications
  • 4.1 About wavelet systems
  • 4.2 Wavelets and signal processing
  • 4.3 Wavelets and fingerprints
  • 4.4 Wavelet packets
  • 4.5 Alternatives to wavelets: Gabor systems
  • 4.6 Exercises
  • 5 Wavelets and their Mathematical Properties
  • 5.1 Wavelets and L2 (?)
  • 5.2 Multiresolution analysis
  • 5.3 The role of the Fourier transform
  • 5.4 The Haar wavelet
  • 5.5 The role of compact support
  • 5.6 Wavelets and singularities
  • 5.7 Best N-term approximation
  • 5.8 Frames
  • 5.9 Gabor systems
  • 5.10 Exercises
  • Appendix A
  • A.1 Definitions and notation
  • A.2 Proof of Weierstrass’ theorem
  • A.3 Proof of Taylor’s theorem
  • A.4 Infinite series
  • A.5 Proof of Theorem 3 7 2
  • Appendix B
  • B.1 Power series
  • B.2 Fourier series for 2?-periodic functions
  • List of Symbols
  • References.

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