An Introduction to Frames and Riesz Bases (Relié)

This completely revised and greatly 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 included 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 reading ; Open problems to stimulate further research.