An Introduction to Frames and Riesz Bases: (Applied and Numerical Harmonic Analysis)

The Applied and Numerical Harmonic Analysis ( ANHA) book series aims to provide the engineering, mathematical, and scientific communities with significant developments in harmonic analysis, ranging from abstract har- monic analysis to basic applications. The title of the series reflects the im- portance of applications and numerical implementation, but richness and relevance of applications and implementation depend fundamentally on the structure and depth of theoretical underpinnings. Thus, from our point of view, the interleaving of theory and applications and their creative symbi- otic evolution is axiomatic. Harmonic analysis is a wellspring of ideas and applicability that has flour- ished, developed, and deepened over time within many disciplines and by means of creative cross-fertilization with diverse areas. The intricate and fundamental relationship between harmonic analysis and fields such as sig- nal processing, partial differential equations (PDEs), and image processing is reflected in our state of the art ANHA series. Our vision of modern harmonic analysis includes mathematical areas such as wavelet theory, Banach algebras, classical Fourier analysis, time- frequency analysis, and fractal geometry, as well as the diverse topics that impinge on them.

Table of Contents:
Preface 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 L2(R) Selected Topics on Gabor Frames Gabor Frames in l2(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 Appendix List of Symbols References Index

Review :
From the reviews: "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." -Zentralblatt Math "The book is a well-written and detailed course into the theory of bases and frames in Hilbert spaces. The composition is very clear, and the proofs are well achieved... In the basic chapters, a large number of carefully chosen examples and exercises are included. That first part can be used in a graduate course. The material of the later chapters is more in the line of current research... I recommend this book to graduate students and researchers working in pure and applied mathematics. It will appeal to an audience interested in the theory behind many signal processing tools stimulating further research." -ZAA "The last decade witnessed a significant change in the field of data representation, with the theory and applications of redundant representations taking center stage and becoming a central research topic in the areas of wavelet and Gabor representations. The specific topic of frame representations received particular attention and became a major theme for these efforts. [This book]...successfully summarizes that progress. Some of its chapters are basic, and are suitable for use in a graduate course in mathematics. Other chapters provide the specialist with a detailed up-to-date review of the state-of-the-art in the field. Other scientists, with more general interest in the area, might use the book as a general reference on the topic." -Journal of Approximation Theory "This is the first book giving a comprehensive overview over the theory of frames and Riesz basis, which has become important in connection with wavelet theory and nonorthogonal signal expansions. Technically speaking frames in a Hilbert space are the correct analogue of a sequence of generators in a finite-dimensional vector space, while the concept of dual frames corresponds to the notion of pseudo-inverse matrices (widely underestimated in standard courses). The book provides a gentle introduction into the field, is suitable for self-study or for the design of a course, and leads from the beginnings to active research areas. Hence it should be found in any library." -Monatshefte fur Mathematik "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)