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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.
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Reviewer: R. Ghrist, University of Illinois, Urbana-Champaign, USA.
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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.