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

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9783319256115: An Introduction to Frames and Riesz Bases (Applied and Numerical Harmonic Analysis)

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 

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About the Author:

Ole Christensen is a Professor at Technical University of Denmark in the Department of Applied Mathematics and Computer Science. His research interests include harmonic analysis, frame expansions, wavelets, and Gabor systems.

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 für 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)

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Buchbeschreibung Springer-Verlag Gmbh Jun 2016, 2016. Buch. Buchzustand: Neu. Neuware - 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 704 pp. Englisch. Buchnummer des Verkäufers 9783319256115

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