Fourier Analysis on Local Fields. (MN-15)

✍ Scribed by M. H. Taibleson

• Publisher: Princeton University Press
• Year: 2015
• Tongue: English
• Leaves: 307
• Series: Mathematical Notes; 15
• Category: Library

✦ Synopsis

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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✦ Table of Contents

Cover
Table of Contents
Preface
Introduction
I. Introduction to Local Fields
II. Fourier Analysis on K, the One-Dimension Case
III. Fourier Analysis on Kn
IV. Regularization and the Theory of Regular and Sub-Regular Functions
V. The Littlewood-Paley Function and Some Applications
VI. Multipliers and Singular Integral Operators
VII. Conjugate Systems of Regular Functions and an F. and M. Riesz Theorem
VIII. Almost Everywhere Convergence of Fourier Series
Bibliography