An Introduction to Frames and Riesz Bases [Hardcover]

Ole Christensen (Author)

Book Description

Publication Date: December 13, 2002 | ISBN-10: 0817642951 | ISBN-13: 978-0817642952 | Edition: 1

The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature.

Editorial Reviews

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)

From the Back Cover

The theory for frames and bases has developed rapidly in recent years because of its role as a mathematical tool in signal and image processing. In this self-contained work, frames and Riesz bases are presented from a functional analytic point of view, emphasizing their mathematical properties. This is the first comprehensive book to focus on the general properties and interplay of frames and Riesz bases, and thus fills a gap in the literature. Key features: * 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 with applications and connections to time-frequency analysis, wavelets, and nonharmonic Fourier series * Selected research topics presented with recommendations for more advanced topics and further reading * Open problems to simulate further research An Introduction to Frames and Riesz Basis 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.

Product Details

• Hardcover: 464 pages
• Publisher: Birkhäuser Boston; 1 edition (December 13, 2002)
• Language: English
• ISBN-10: 0817642951
• ISBN-13: 978-0817642952
• Product Dimensions: 9.3 x 6.3 x 1.1 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 4.5 out of 5 stars  See all reviews (2 customer reviews)
• Amazon Best Sellers Rank: #1,273,470 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

8 of 8 people found the following review helpful:

5.0 out of 5 stars A new tool!, February 11, 2003

By 

Palle E T Jorgensen "Palle Jorgensen" (Iowa City, Iowa United States) - See all my reviews
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This review is from: An Introduction to Frames and Riesz Bases (Hardcover)

The subject of the book is a new tool in math, with a host of exciting applications. Of course, the subject has roots in classical ideas from harmonic analysis. But the book covers an explosive and exciting variety of developments since roughly 1990, and it is presented in the form of a graduate text. The basic idea begins with linear algebra, and progresses to expansions in function spaces, and multiresolutions. It will be useful to anyone who wants to learn from scratch about the underlying principles, the new results, and the applications. It is well written. A student of mine picked it up accidentally from my desk, and couldn't put it down. After awhile, he had completely forgotten what he came to see me about. It could have been some of the applications, such as antenna theory, wavelets, time-frequency analysis, uses in Radar, speech processing...

4.0 out of 5 stars Good intermediate level introduction to Frames, November 12, 2011

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Manchor Ko - See all my reviews
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This review is from: An Introduction to Frames and Riesz Bases (Hardcover)

The book cover the basic theories of frames well and is not overburdened with proofs. The first 5-6 chapters flows well and does not require a lot of background besides linear algebra.

The best chapters for me personally is ch.13 and ch.16 - "Frames and Multiresolution Analysis", "Approximation of the Inverse Frame Operator". While it does not cover some of the modern efficient methods for frames with specialized structures it covered the important case of a general frame.

I wish the book treated tight frames a little more. Besides their practical application, tight frames give us a good perspective on what we are gaining and losing from the general frame to a tight frame if we focus on the Parseval equation as well as what the inverse frame operator is like for both.

Interested readers should consider the excellent textbook "Frames for Undergraduates" also. It covers tight frames a little better and give intuition as to why they are important by focusing on the Parseval property. Frames for Undergraduates (Student Mathematical Library)

I will be giving a talk on "Frames, Sparsity and Global Illumination" at GDC 2012.