An Introduction to Frames and Riesz Bases

Description

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  Front Matter....Pages i-xxv Frames in Finite-Dimensional Inner Product Spaces....Pages 1-46 Infinite-Dimensional Vector Spaces and Sequences....Pages 47-66 Bases....Pages 67-108 Bases and Their Limitations....Pages 109-118 Frames in Hilbert Spaces....Pages 119-151 Tight Frames and Dual Frame Pairs....Pages 153-164 Frames Versus Riesz Bases....Pages 165-181 Selected Topics in Frame Theory....Pages 183-198 Frames of Translates....Pages 199-240 Gabor Frames and Duality....Pages 241-256 Selected Topics on Gabor Frames....Pages 257-286 Frame Multiresolution Analysis....Pages 287-325 Wavelet Frames via Extension Principles....Pages 327-357 Selected Topics on Wavelet Frames....Pages 359-384 Frames on Locally Compact Abelian Groups....Pages 385-405 Perturbation of Frames....Pages 407-415 Approximation of the Inverse Frame Operator....Pages 417-444 Expansions in Banach Spaces....Pages 445-478 Back Matter....Pages 479-492 ....Pages 493-517