An introduction to frames and Riesz bases. (English) Zbl 1017.42022

Christensen, Ole

Applied and Numerical Harmonic Analysis. Boston, MA: Birkhäuser. xx, 440 p. (2003).

The aim of this book is to present parts of the modern theory of bases and frames in Hilbert spaces in a way that the material can be used in a graduate course, as well as by professional readers. For use in a graduate course, a number of exercises are included at the end of each chapter. The number of exercises give a hint to the level of the chapter: there are many exercises in the introductory chapters, but only a few in the advanced chapters. In the same spirit, almost all the results in the introductory chapters appear with full proofs, but some proofs are skipped in later chapters. The book is well written, the proofs are clear and not too terse, and the work is well suited for use as a textbook. The author has made many contributions to the theory of frames and Riesz bases, and the book benefits from his scope and perspective. The chapter titles are: Frames in finite–dimensional inner product spaces. Infinite–dimensional vector spaces and sequences. Bases. Bases and their limitations. Frames in Hilbert spaces. Frames versus Riesz bases. Frames of translates. Gabor frames in \(L^2(R)\). Selected topics on Gabor frames. Gabor frames in \(\ell^2(Z)\). General wavelet frames. Dyadic wavelet frames. Frame multiresolution analysis. Wavelet frames via extension principles. Perturbation of frames. Approximation of the inverse frame operator. Expansions in Banach spaces. There is also an appendix where some mathematical prerequisites are briefly discussed, and a list of symbols. The bibliography lists 294 items.

Reviewer: Richard A.Zalik (Auburn University)

MSC:

42C40	Nontrigonometric harmonic analysis involving wavelets and other special systems
42-02	Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces
42C15	General harmonic expansions, frames
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)

Keywords:

frames; bases; Riesz bases; Hilbert spaces; frames of translates; wavelets; dyadic wavelet frames; multiresolution analysis; frame multiresolution analysis; Gabor frames; frame perturbations; Banach spaces