An Introduction to Frames and Riesz Bases / by Ole Christensen.

Format:

Book

Author/Creator:

Christensen, Ole., Author.

Series:

Applied and Numerical Harmonic Analysis, 2296-5009

Language:

English

Subjects (All):

Functional analysis.

Harmonic analysis.

Operator theory.

Signal processing.

Image processing.

Speech processing systems.

Functional Analysis.

Abstract Harmonic Analysis.

Operator Theory.

Signal, Image and Speech Processing.

Local Subjects:

Functional Analysis.

Abstract Harmonic Analysis.

Operator Theory.

Signal, Image and Speech Processing.

Physical Description:

1 online resource (XXV, 704 p. 17 illus., 5 illus. in color.)

Edition:

2nd ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Birkhäuser, 2016.

Summary:

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 An Introduction to Frames and Riesz Bases 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 An Introduction to Frames and Riesz Bases 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 .

Contents:

Frames in Finite-dimensional Inner Product Spaces

Infinite-dimensional Vector Spaces and Sequences

Bases

Bases and their Limitations

Frames in Hilbert Spaces

Tight Frames and Dual Frame Pairs

Frames versus Riesz Bases

Selected Topics in Frame Theory

Frames of Translates

Shift-Invariant Systems in l2(R)

Gabor Frames in L2(R)

Gabor Frames and Duality

Selected Topics on Gabor Frames

Gabor Frames in ℓ2(Z),L2(0,L),CL

General Wavelet Frames in L2(R)

Dyadic Wavelet Frames for L2(R)

Frame Multiresolution Analysis

Wavelet Frames via Extension Principles

Selected Topics on Wavelet Frames

Generalized Shift-Invariant Systems in L2(Rd)

Frames on Locally Compact Abelian Groups

Perturbation of Frames

Approximation of the Inverse Frame Operator

Expansions in Banach Spaces. Appendix.

Notes:

Description based on publisher supplied metadata and other sources.

ISBN:

3-319-25613-0

OCLC:

1076230993