Distributions and Fourier Transforms

William F. Donoghue, "Distributions and Fourier Transforms"
Academic Pr | 1969-06 | ISBN: 0122206509 | 327 pages | PDF | 10,8 MB

In this book I try to give a readable introduction to the modern theory of
the Fourier transform and to show some interesting applications of that
theory in higher analysis. The book is directed to students having only a
moderate preparation in real and complex analysis. More exactly, I suppose
the reader to be familiar with the elements of real variables and Lebesgue
integration and to have some knowledge of analytic functions. Further along
in the book both Hilbert spaces and LP-spaces play a role, but the reader is
presumed to know only a little about either topic, much less, in fact, than
appears in any standard modern real variable textbook.
Much of the material the student is expected to know is reviewed in the
first part of the book, which also serves to establish our 'conventions of
notation and terminology. Some topics from advanced calculus and analytic
function theory are treated here. There have also been adjoined brief discussions
of linear topological spaces, analytic functions of several variables,
as well as certain aspects of convexity; these subjects are perhaps not strictly
needed for the study of the Fourier transform as we undertake it.
Not everything in Part I is needed for the study of Part I1 which presents
the theory of distributions on the n-dimensional real space as well as the
theory of the Fourier transform for temperate distributions. The machinery
developed in Part 11 makes it possible to obtain significant results in harmonic
analysis in a fairly simple and direct way; this is done in Part 111. The whole
book can be covered conveniently in a one-year course if one or two special
topics in the third part are omitted.
Much of the book closely follows the lectures in harmonic analysis
given by L. Hormander at Stockholm University during the academic year
1958-1959. However, a number of topics covered in those lectures have
been omitted, while a good deal of potential theory and analytic function
theory has been adjoined; it would be surprising if Professor Hormander
cared to acknowledge the result as his own. Nevertheless, almost everything
in this book has been taught me by L. Hormander and N. Aronszajn.
There are certain inconsistencies in the presentation. To make the book
accessible to as wide a readership as possible I have avoided the treatment of
distributions on manifolds and never refer to an exterior differential form.
This has made it desirable to accept the Green's formula without proof,
although it is only needed here for spheres. Sometimes a theorem is proved
with the tacit assumption that the functions or linear spaces occurring in the
argument are all real, and later that theorem is invoked in a context where
the scalars are complex. This abuse is preferred to the repetition of some
incantation assuring the reader that the arguments may be modified to cover
the case of complex scalars. I have tried to make the notations as traditional
and natural as possible, but have not been able to avoid some trivial ambiguities.
Thus, for example, a system of points in R" is generally written x k ,
although the same notation is used for the coordinate functions themselves.
A book covering such a wide range of material is bound to contain mistakes.
These, I think, are unimportant, so long as the book conveys the
mathematical spirit of the apostolic, nay, the Petrine succession, extending
from Gauss, Riemann, and Dirichlet, through Hilbert, Courant, Friedrichs,
and John.

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