A revised and expanded second edition of Reiter's classic text Classical Harmonic Analysis and Locally Compact Groups (Clarendon Press 1968). It deals with various developments in analysis centring around around the fundamental work of Wiener, Carleman, and especially A. Weil. It starts with the classical theory of Fourier transforms in euclidean space, continues with a study at certain general function algebras, and then discusses functions defined on locally compact groups. The aim is, firstly, to bring out clearly the relations between classical analysis and group theory , and secondly, to study basic properties of functions on abelian and non-abelian groups. The book gives a systematic introduction to these topics and endeavours to provide tools for further research. In the new edition relevant material is added that was not yet available at the time of the first edition.
Author(s): the late Hans Reiter, Jan D. Stegeman
Series: London Mathematical Society Monographs New Series
Edition: 2
Publisher: Oxford University Press, USA
Year: 2001
Language: English
Pages: 337
Contents......Page 6
Reader's guide......Page 8
1.1 The algebra L^1(\R^\nu); Fourier transforms......Page 11
1.2 Functions in L^1(\R^\nu)......Page 17
1.3 The Wiener-Lévy theorem......Page 20
1.4 Wiener's theorem......Page 22
1.5 Some subalgebras of L^1(\R^\nu)......Page 26
1.6 The algebras L^1_\alpha(\R^\nu); Beurling algebras......Page 28
2.1 Function algebras......Page 35
2.2 Topological function algebras and Wiener's theorem......Page 38
2.3 Examples of topological function algebras......Page 41
2.4 The generalization of Wiener's theorem......Page 45
2.5 Ditkin sets for function algebras......Page 50
2.6 Ideals in function algebras; examples......Page 54
2.7 Sets of spectral synthesis; examples, counter-examples......Page 59
3.1 Locally compact groups......Page 66
3.2 Integration on locally compact spaces......Page 71
3.3 The Haar measure......Page 92
3.4 L^1-spaces on groups......Page 107
3.5 The algebra L^1(G)......Page 114
3.6 The space L^\infty(G)......Page 123
3.7 Beurling algebras......Page 129
4.1 Locally compact abelian groups......Page 138
4.2 Duality theory and structure theory......Page 142
4.3 Some examples......Page 148
4.4 The foundations of harmonic analysis......Page 153
4.5 The principle of relativization......Page 160
5.1 The functions \sigma and \tau......Page 162
5.2 Lemmas on functions in L^1(G)......Page 165
5.3 Lemmas on functions in L^1(G) (continued)......Page 169
5.4 Some properties of L^1(G) and \F^1(\hat{G})......Page 170
5.5 Poisson's formula......Page 174
6.1 Wiener's theorem for groups......Page 179
6.2 Segal algebras, the abelian case......Page 183
6.3 Beurling algebras......Page 194
7.1 Definition and properties of the spectrum......Page 204
7.2 Relativization and the spectrum......Page 210
7.3 Sets of spectral synthesis for Beurling algebras......Page 213
7.4 Ditkin sets on groups......Page 222
8.1 Quasi-invariant measures on quotient spaces......Page 231
8.2 The space L^1(G/H)......Page 240
8.3 The property P_p......Page 244
8.4 The invariant convex hull of a function in L^1(G)......Page 252
8.5 The property (\M)......Page 257
8.6 The equivalence of the properties (\M) and P_1......Page 261
8.7 The structure of groups with the property P_1......Page 266
A.1 Domar's theorem......Page 272
A.2 Malliavin's theorem......Page 277
A.3 More on Segal algebras......Page 282
A.4 The difference spectrum......Page 292
B Notes and additional references......Page 303
References......Page 312
Summary of notations......Page 326
Subject index......Page 331
