Introduction to Abstract Harmonic Analysis

Lynn H. Loomis

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486282312
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This classic monograph is the work of a prominent contributor to the field of harmonic analysis. Geared toward advanced undergraduates and graduate students, it focuses on methods related to Gelfand's theory of Banach algebra. Prerequisites include a knowledge of the concepts of elementary modern algebra and of metric space topology.
The first three chapters feature concise, self-contained treatments of measure theory, general topology, and Banach space theory that will assist students in their grasp of subsequent material. An in-depth exposition of Banach algebra follows, along with examinations of the Haar integral and the deduction of the standard theory of harmonic analysis on locally compact Abelian groups and compact groups. Additional topics include positive definite functions and the generalized Plancherel theorem, the Wiener Tauberian theorem and the Pontriagin duality theorem, representation theory, and the theory of almost periodic functions.

Chapter I
______________

TOPOLOGY

§ 1. SETS

1A. The reader is assumed to be familiar with the algebra of sets, and the present paragraph is confined to notation. The curly bracket notation is used to name specific sets. Thus {a, b} is the set whose elements are a and b, and {a₁ ···, a_n} is the set whose elements are a₁ ···, a_n. However, sets cannot generally be named by listing their elements, and more often are defined by properties, the notation being that {x: ( )} is the set of all x such that ( ). We write “a

A” for “a is a member of A” and B ⊂ A” for “B is a subset of A”; thus A = {x: x

A}.

The ordinary “cup” and “cap” notation will be used for combinations of sets: A ∪ B for the union of A and B, and A ∩ B for their intersection;

for the union of the sets of the countable family {A_n}, and

for their intersection. More generally, if

is any family of sets, then

or

is the union of the sets in

, and

is their intersection. The complement of A, designated A′, is understood to be taken with respect to the “space” in question, that is, with respect to whatever class is the domain of the given discussion. The null class is denoted by ∅. The only symbols from logic which will be used are “⇒” for “if ··· then” and “⇔” for “if and only if.”

1B. A binary relation “<” between elements of a class A is called a partial ordering of A (in the weak sense) if it is transitive (a < b and b < c ⇒ a < c), reflexive (a < a for every a

A) and if a < b and b < a ⇒ a = b. A is called a directed set under “<” if it is partially ordered by “&...

Table of contents

• Cover Page
• Title Page
• CHAPTER I: TOPOLOGY
• CHAPTER II: BANACH SPACES
• CHAPTER III: INTEGRATION
• CHAPTER IV: BANACH ALGEBRAS
• CHAPTER V: SOME SPECIAL BANACH ALGEBRAS
• CHAPTER VI: THE HAAR INTEGRAL
• CHAPTER VII: LOCALLY COMPACT ABELIAN GROUPS
• CHAPTER VIII: COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS
• CHAPTER IX: SOME FURTHER DEVELOPMENTS
• Bibliography
• Index
• Back Cover