An introduction to frames and Riesz bases
type: Item
instance of: scholarly article
title: An introduction to frames and Riesz bases
zbMATH Open document ID: 1017.42022
author: Ole Christensen
published in: Applied and Numerical Harmonic Analysis
publication date: 2003-03-19T00:00:00Z
review text: 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.
reviewed by: Richard A. Zalik
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 42-02
Mathematics Subject Classification ID: 42C15
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46C05
zbMATH DE Number: 1885134
zbMATH Keywords: frames
zbMATH Keywords: bases
zbMATH Keywords: Riesz bases
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: frames of translates
zbMATH Keywords: wavelets
zbMATH Keywords: dyadic wavelet frames
zbMATH Keywords: multiresolution analysis
zbMATH Keywords: frame multiresolution analysis
zbMATH Keywords: Gabor frames
zbMATH Keywords: frame perturbations
zbMATH Keywords: Banach spaces
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1885134