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William F. Donoghue, "Distributions and Fourier Transforms"

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.

March, 1969 WILLIAMF. DONOGHUJRE.,

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