Approximation Theory From Taylor Polynomials to Wavelets /

• Κύριοι συγγραφείς: Christensen, Ole (Συγγραφέας), Christensen, Khadija L. (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Boston, MA : Birkhäuser Boston : Imprint: Birkhäuser, 2005.
• Σειρά: Applied and Numerical Harmonic Analysis
• Θέματα: Mathematics.; Harmonic analysis.; Approximation theory.; Fourier analysis.; Functional analysis.; Applied mathematics.; Engineering mathematics.; Fourier Analysis.; Approximations and Expansions.; Abstract Harmonic Analysis.; Functional Analysis.; Applications of Mathematics.; Signal, Image and Speech Processing.

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...

Πίνακας περιεχομένων:

• 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.