Customer Reviews

An Introduction to Frames and Riesz Bases

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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...

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.