This book presents a development of the basic facts about harmonic
analysis on local fields and the n-dimensional vector spaces over
these fields. It focuses almost exclusively on the analogy between the
local field and Euclidean cases, with respect to the form of statements,
the manner of proof, and the variety of applications.
The force of the analogy between the local field and Euclidean cases
rests in the relationship of the field structures that underlie the
respective cases. A complete classification of locally compact,
non-discrete fields gives us two examples of connected fields (real and
complex numbers); the rest are local fields (p-adic numbers, p-series
fields, and their algebraic extensions). The local fields are studied in
an effort to extend knowledge of the reals and complexes as locally
compact fields.
The author's central aim has been to present the basic facts of Fourier
analysis on local fields in an accessible form and in the same spirit as
in Zygmund's Trigonometric Series (Cambridge, 1968) and in
Introduction to Fourier Analysis on Euclidean Spaces by Stein and
Weiss (1971).
Originally published in 1975.
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