Advances in Pseudo-Differential Operators

By Ryuichi Ashino, Paolo Boggiatto, Man-Wah Wong (editors)

Advances in Pseudo-Differential Operators

Ryuichi Ashino, Paolo Boggiatto, Man-Wah Wong (editors)
Birkhauser (Operator Theory: Advances and Applications), (2004) 233 pp., price USD 139.-
ISBN: 3764371404.
Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.
Level: advanced.

The volume under review serves as a proceedings for the special session on pseudo-differential operators given at the Fourth Congress of the International Society for Analysis, Applications and Computation (Toronto, August 2003). The subject material is pseudo-differential operators, with an emphasis on applications to partial differential equations, wavelets, and image processing.

Roughly speaking, microlocal analysis comprises those tools in harmonic analysis which followed the classical Fourier theory as relates to, e.g., distributions and Sobolev spaces, among which are included pseudo-differential operators and generalized Fourier integral operators. Such techniques are increasingly important and effective in non-elliptic PDEs. The technicalities in this subject are substantial, and the present volume reflects this aspect of the theory.

The book is fairly broad in scope (for a specialized text on pseudo-differential operators, that is). There is a good mix between hard analysis and applications. It is, however, a coarse mix: chapters switch rather discontinuously from the pure to the applied, without much of a storyline. This is neither a surprise nor a disappointment given the text's origins as a conference proceedings. The reader should be warned, however, that this is a book for specialized rather than cover-to-cover reading. The level of the book is advanced, and the casual reader who is not familiar with the language of harmonic analysis and Sobolev spaces may find the text challenging.

There are twelve chapters with a prefatory chapter on microlocal analysis and its applications. The chapters likely to be of most interest to readers of this review include: the first [overview] chapter and the final three chapters on [respectively] microlocal analysis with applications to image processing; wavelets and system identification; and wavelet bases for partial differential operators with applications to thin films.

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