A Guide to Distribution Theory and Fourier Transforms - ISBN:0849382734

A Guide to Distribution Theory and Fourier Transforms

Date: 22 April 2011, 12:32

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This book provides a concise exposition of the basic ideas of the theory of distribution and Fourier transforms and its application to partial differential equations. The author clearly presents the ideas, precise statements of theorems, and explanations of ideas behind the proofs. Methods in which techniques are used in applications are illustrated, and many problems are included. The book also introduces several significant recent topics, including pseudo-differential operators, wave front sets, wavelets, and quasicrystals. Background mathematical prerequisites have been kept to a minimum, with only a knowledge of multidimensional calculus and basic complex variables needed to fully understand the concepts in the book.
in applied analysis and mathematical physics.
Summary: Excellent
Rating: 5
Distributions are objects most physicists will frequently encounter during their career, but, surprinsingly, the subject is not given the place it deserves in the current ordinary science curriculum.
I would particularly recommend this book to physics students willing to learn the foundation of distribution theory and its close ties to Fourier transforms. Distribution theory is, basically speaking, a way of making rigorous the operations physicists find Ok to carry on functions, that otherwise wouldn't rigorously make sense. Distribution theory therefore provides a useful way of checking, in the process of a calculation, if it is allowed (according to the extended rules of distribution theory), or if it is definitely dubious (e.g. current distribution theory doesn't provide a mean of making sense of a product of Dirac delta functions, while such expressions sometimes pop out in the context of quantum field theory ; nevertheless, there exist other formal theories, such as Colombo calculus that aim at justifying this ; yet, for some reason, they seem to bear less power than the original distribution theory).
This work is an easy, gentle, pedagogical piece of mathematical exposition.
The subject is wonderfully motivated.
As such, this book is suited to self-study.
It could also be used as a textbook for an introductory course on the subject, or as an introductory reading to more advanced texts (Aizenman, for instance).
Highly recommended.

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