Approximation Theory

From Taylor Polynomials to Wavelets
6th 2004., Corr. 2. Printing 2005, Corr. 3rd Printing 2006 Auflage
ISBN: 978-0-8176-3600-5
Verlag: Birkhauser Boston

Buch, Englisch, 156 Seiten, Format (B × H): 156 mm x 235 mm, Gewicht: 307 g

Reihe: Applied and Numerical Harmonic Analysis

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 and the influence of classical disciplines on many areas of modern mathematics and applications.  Featuring classical, illustrative examples and constructions, exercises, and a discussion of the role of wavelets to areas such as digital signal processing and data compression, the book is one of the few to describe wavelets in words rather than mathematical symbols.

Zielgruppe

Graduate

Autoren/Hrsg.

Christensen, Ole

Christensen, Khadija Laghrida

Fachgebiete

• Technische Wissenschaften Elektronik | Nachrichtentechnik Nachrichten- und Kommunikationstechnik
• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Technische Wissenschaften Sonstige Technologien | Angewandte Technik Signalverarbeitung, Bildverarbeitung, Scanning
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen

Inhaltsverzeichnis

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.

Über Autor(innen)

Ole Christensen is the author of (0-8176-4295-1).