An Introduction to Frames and Riesz Bases

An Introduction to Frames and Riesz Bases
An Introduction to Frames and Riesz Bases
Birkhäuser | Mathematics | June 25, 2016 | ISBN-10: 3319256114 | 704 pages | pdf | 8.74 mb

Authors: Christensen, Ole
Extensive and updated exercises included at the end of every chapter
Selected research topics presented with recommendations for more advanced topics and further reading
Basic results presented in an accessible way for both pure and applied mathematicians
Full proofs included in introductory chapters; only basic knowledge of functional analysis required

This revised and expanded monograph presents the general theory for frames and Riesz bases in Hilbert spaces as well as its concrete realizations within Gabor analysis, wavelet analysis, and generalized shift-invariant systems. Compared with the first edition, more emphasis is put on explicit constructions with attractive properties. Based on the exiting development of frame theory over the last decade, this second edition now includes new sections on the rapidly growing fields of LCA groups, generalized shift-invariant systems, duality theory for as well Gabor frames as wavelet frames, and open problems in the field.

Key features include:
*Elementary introduction to frame theory in finite-dimensional spaces
* Basic results presented in an accessible way for both pure and applied mathematicians
* Extensive exercises make the work suitable as a textbook for use in graduate courses
* Full proofs includ
ed in introductory chapters; only basic knowledge of functional analysis required
* Explicit constructions of frames and dual pairs of frames, with applications and connections to time-frequency analysis, wavelets, and generalized shift-invariant systems
* Discussion of frames on LCA groups and the concrete realizations in terms of Gabor systems on the elementary groups; connections to sampling theory
* Selected research topics presented with recommendations for more advanced topics and further readin
g
* Open problems to stimulate further research

will be of interest to graduate students and researchers working in pure and applied mathematics, mathematical physics, and engineering. Professionals working in digital signal processing who wish to understand the theory behind many modern signal processing tools may also find this book a useful self-study reference.

Review of the first edition:
"Ole Christensen's is a first-rate introduction to the field ... . The book provides an excellent exposition of these topics. The material is broad enough to pique the interest of many readers, the included exercises supply some interesting challenges, and the coverage provides enough background for those new to the subject to begin conducting original research."
-Eric S. Weber, American Mathematical Monthly, Vol. 112, February, 2005

Number of Illustrations and Tables
12 b/w illustrations, 5 illustrations in colour
Topics
Functional Analysis
Abstract Harmonic Analysis
Operator Theory
Signal, Image and Speech Processing

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