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Title: The Schur Algorithm, Reproducing Kernel Spaces and System Theory by Daniel Alpay, Stephen S. Wilson ISBN: 0-8218-2155-5 Publisher: American Mathematical Society Pub. Date: 01 August, 2001 Format: Mass Market Paperback Volumes: 1 List Price(USD): $49.00

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Average Customer Rating: 5 (1 review)

Rating: 5
Summary: An Impressive Survey
Comment: This book is an impressive survey of recent research in the applications of Reproducing Kernel Hilbert Spaces to complex analytic function theory, and, in particular to the study of functions which are analytic on the open unit disk in the complex plane.

This book is accessible to anybody who has mastered the essentials of complex function theory, banach space theory, and hilbert space theory which are found in Walter Rudin's book "Real and Complex Analysis". You don't need even the advanced topics in complex function theory in Rudin's book to understand Alpay's book. Hence, this book is excellent for graduate students who want to do research in this field.

If this book accomplished only that, it would be impressive enough, but it does much more.

At the same time that this book surveys current research (current as of 2000) in the applications of RKHS to complex function theory, this book surveys the applications of RKHS to mathematics as a whole! Very little is left out, and the reader is given short intuitive explanations of just how each application works. It is amazing how the references to a vast array of RKHS applications are salt and peppered throughout the first half of this book in a way which fits in completely naturally with the over-all flow of the argument. Professor Alpay makes it all so easy to grasp because of his clear and straightforward writing style. Of course, credit must be give to the translator Stephen Wilson in this regard too.

Professor Alpay's excellent bibliography is a little weak in the area of references to applications of RKHS to signal processing and to data mining. I would like to suggest three references which, when combined with Alpay's references, give an even more excellent bibliography.

First is the papers in "Reproducing Kernel Hilbert Spaces: Applications to Statistical Signal Processing" and especialy the four papers by Kailath labelled "Parts I, II, III, and IV". This is available on Amazon.com.

Second is "Learning Kernel Classifiers" by Herbrich for the applications to data mining. This is also available on amazon.com.

Third is "Detection, Estimation, and Modulation Theory" by Harry Van Trees Part I, especially page 368 equation 434 and page 297. Van Trees' book is still the best for the nuts and bolts of the application of RKHS to signal processing as far as I know. For example, Professor Van Trees gives an excellent explanation of how the theory can be extended (almost beyond the breaking point) to handle the white noise case.

Combining Van Trees insight into the applications with Kailath's insight into the theory is a powerful combination. Herbrich shows how these powerful insights are being leveraged today in the field of data mining and especially in the field of support vector machines and their extensions to kernel based methods in general.

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