Toeplitz Forms and Their Applications (AMS Chelsea Publishing) Review

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(More customer reviews)I think that my previous reviewer has been a bit unfair with this work. It was written in the mid-50s and it's still useful. Its main purpose is to develop techniques to deal with the computation of the eigenvalues of Toeplitz matrices. The first chapters are more analytically-oriented, but the treatment of the topics of functional analysis and approximation theory that are necessary to understand the rest of the work is superb. Of course, if you are a mathematician like me you'll be aware of the spectacular development of functional analysis over the last 50 years so you'll have little trouble with it. The rest of the work is devoted to the application of Toeplitz forms to time-series analysis. Although spectral analysis wasn't very common in econometric applications (try to carry out a manual computation of a periodogram and you'll understand why), computer programs have changed dramatically this situation, so it's being increasingly used for the study of business cycles. But computer results without a "solid" mathematical foundation to understand its meaning are nothing, and this is precisely what this book offers the reader.

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From the Preface (1955): The first part of the present exposition is devoted to the theory of Toeplitz forms. The second part deals with applications, in particular to the calculus of probability and mathematical statistics. Neither part claims completeness in any way. Our purpose has been to elucidate the principal ideas of this remarkable chapter of modern analysis and to help the interested student of mathematical statistics to acquire a working knowledge of the subject. The somewhat protracted Chapter 1 explains not only the notation employed but contains also the definition of important auxiliary concepts and the exposition of basic results which will be used later. This arrangement avoids interruptions in the main text ... [It is assumed] that the reader is in possession of the fundamental facts of the theory of functions. ``In chapters 2 and 3 certain topics appear which were treated in the book on orthogonal polynomials by G. Szego. In view of the progress made in this subject since the publication of that book (1939) it was possible to bring some details in an improved setting. The other chapters contain partly old and partly more recent results, some older facts in a new setting, and finally some completely new results. Chapter 1-6 and Chapter 9 have been prepared by Szego, the other chapters by Grenander ... ''

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