A Course in Abstract Harmonic Analysis
1st Edition
Published on 27. December 1994
Book
Hardback
290 pages
978-0-8493-8490-5 (ISBN)
Description
Abstract theory remains an indispensable foundation for the study of concrete cases. It shows what the general picture should look like and provides results that are useful again and again. Despite this, however, there are few, if any introductory texts that present a unified picture of the general abstract theory.
A Course in Abstract Harmonic Analysis offers a concise, readable introduction to Fourier analysis on groups and unitary representation theory. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book proceeds to the basic facts about locally compact groups, Haar measure, and unitary representations, including the Gelfand-Raikov existence theorem. The author devotes two chapters to analysis on Abelian groups and compact groups, then explores induced representations, featuring the imprimitivity theorem and its applications. The book concludes with an informal discussion of some further aspects of the representation theory of non-compact, non-Abelian groups.
A Course in Abstract Harmonic Analysis offers a concise, readable introduction to Fourier analysis on groups and unitary representation theory. After a brief review of the relevant parts of Banach algebra theory and spectral theory, the book proceeds to the basic facts about locally compact groups, Haar measure, and unitary representations, including the Gelfand-Raikov existence theorem. The author devotes two chapters to analysis on Abelian groups and compact groups, then explores induced representations, featuring the imprimitivity theorem and its applications. The book concludes with an informal discussion of some further aspects of the representation theory of non-compact, non-Abelian groups.
More details
Language
English
Place of publication
Bosa Roca
United States
Publishing group
Taylor & Francis Inc
Target group
College/higher education
Professional and scholarly
Illustrations
750 equations
Dimensions
Height: 234 mm
Width: 159 mm
Weight
600 gr
ISBN-13
978-0-8493-8490-5 (9780849384905)
Schweitzer Classification
Other editions
New editions
Gerald B. Folland
A Course in Abstract Harmonic Analysis
Book
09/2015
2nd Edition
Chapman & Hall/CRC
€133.00
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Content
Banach Algebras and Spectral Theory
Banach Algebras: Basic Concepts
Gelfand Theory
Nonunital Banach Algebras
The Spectral Theorem
Spectral Theory of *-Representations
Notes and References
Locally Compact Groups
Topological Groups
Haar Measure
Interlude: Some Technicalities
The Modular Function
Convolutions
Homogeneous Spaces
Notes and References
Basic Representation Theory
Unitary Representations
Representations of a Group and its Group Algebra
Functions of Positive Type
Notes and References
Analysis on Locally Compact Abelian Groups
The Dual Group
The Fourier Transform
The Pontrjagin Duality Theorem
Representations of Locally Compact Abelian Groups
Closed Ideals in L1(G)
Spectral Synthesis
The Bohr Compactification
Notes and References
Analysis on Compact Groups
Representations of Compact Groups
The Peter-Weyl Theorem
The Fourier Transform on Compact Groups
Examples
Notes and References
Induced Representations
The Inducing Construction
The Frobenius Reciprocity Theorem
Pseudomeasures and Induction in Stages
Systems of Imprimitivity
The Imprimitivity Theorem
Introduction to the Mackey Machine
Examples
Notes and References
Further Topics in Representation Theory
The Group C* Algebra
The Structure of the Dual Space
Tensor Products
Direct Integral Decompositions
The Plancherel Theorem
Examples
Appendix 1. Hilbert Space Miscellany
Appendix 2. Trace-Class and Hilbert-Schmidt Operators
Appendix 3. Vector-Valued Integrals
Bibliography
Index
Banach Algebras: Basic Concepts
Gelfand Theory
Nonunital Banach Algebras
The Spectral Theorem
Spectral Theory of *-Representations
Notes and References
Locally Compact Groups
Topological Groups
Haar Measure
Interlude: Some Technicalities
The Modular Function
Convolutions
Homogeneous Spaces
Notes and References
Basic Representation Theory
Unitary Representations
Representations of a Group and its Group Algebra
Functions of Positive Type
Notes and References
Analysis on Locally Compact Abelian Groups
The Dual Group
The Fourier Transform
The Pontrjagin Duality Theorem
Representations of Locally Compact Abelian Groups
Closed Ideals in L1(G)
Spectral Synthesis
The Bohr Compactification
Notes and References
Analysis on Compact Groups
Representations of Compact Groups
The Peter-Weyl Theorem
The Fourier Transform on Compact Groups
Examples
Notes and References
Induced Representations
The Inducing Construction
The Frobenius Reciprocity Theorem
Pseudomeasures and Induction in Stages
Systems of Imprimitivity
The Imprimitivity Theorem
Introduction to the Mackey Machine
Examples
Notes and References
Further Topics in Representation Theory
The Group C* Algebra
The Structure of the Dual Space
Tensor Products
Direct Integral Decompositions
The Plancherel Theorem
Examples
Appendix 1. Hilbert Space Miscellany
Appendix 2. Trace-Class and Hilbert-Schmidt Operators
Appendix 3. Vector-Valued Integrals
Bibliography
Index