A Course in Abstract Harmonic Analysis: 2nd Edition (Hardback) book cover

A Course in Abstract Harmonic Analysis

2nd Edition

By Gerald B. Folland

Chapman and Hall/CRC

305 pages | 3 B/W Illus.

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pub: 2015-09-25
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A Course in Abstract Harmonic Analysis is an introduction to that part of analysis on locally compact groups that can be done with minimal assumptions on the nature of the group. As a generalization of classical Fourier analysis, this abstract theory creates a foundation for a great deal of modern analysis, and it contains a number of elegant results and techniques that are of interest in their own right.

This book develops the abstract theory along with a well-chosen selection of concrete examples that exemplify the results and show the breadth of their applicability. After a preliminary chapter containing the necessary background material on Banach algebras and spectral theory, the text sets out the general theory of locally compact groups and their unitary representations, followed by a development of the more specific theory of analysis on Abelian groups and compact groups. There is an extensive chapter on the theory of induced representations and its applications, and the book concludes with a more informal exposition on the theory of representations of non-Abelian, non-compact groups.

Featuring extensive updates and new examples, the Second Edition:

  • Adds a short section on von Neumann algebras
  • Includes Mark Kac’s simple proof of a restricted form of Wiener’s theorem
  • Explains the relation between SU(2) and SO(3) in terms of quaternions, an elegant method that brings SO(4) into the picture with little effort
  • Discusses representations of the discrete Heisenberg group and its central quotients, illustrating the Mackey machine for regular semi-direct products and the pathological phenomena for nonregular ones

A Course in Abstract Harmonic Analysis, Second Edition serves as an entrée to advanced mathematics, presenting the essentials of harmonic analysis on locally compact groups in a concise and accessible form.


Praise for the Previous Edition

"This delightful book fills a long-standing gap in the literature on abstract harmonic analysis. … To the reviewer's knowledge, no one existing book contains all of the topics that are treated in this one. … [The author's] respect for the subject shows on every hand…through his careful writing style, which is concise, yet simple and elegant. The reviewer would encourage anyone with an interest in harmonic analysis to have this book in his or her personal library. … a fine book that the reviewer would have been proud to write."

—Robert S. Doran in Mathematical Reviews®, Issue 98c

Table of Contents

Banach Algebras and Spectral Theory

Banach Algebras: Basic Concepts

Gelfand Theory

Nonunital Banach Algebras

The Spectral Theorem

Spectral Theory of ∗-Representations

Von Neumann Algebras

Notes and References

Locally Compact Groups

Topological Groups

Haar Measure

Interlude: Some Technicalities

The Modular Function


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

Fourier Analysis on Compact Groups


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: The Classics

More Examples, Good and Bad

Notes and References

Further Topics in Representation Theory

The Group C* Algebra

The Structure of the Dual Space

Tensor Products of Representations

Direct Integral Decompositions

The Plancherel Theorem



A Hilbert Space Miscellany

Trace-Class and Hilbert-Schmidt Operators

Tensor Products of Hilbert Spaces

Vector-Valued Integrals

About the Author

Gerald B. Folland received his Ph.D in mathematics from Princeton University, New Jersey, USA in 1971. After two years at the Courant Institute of Mathematical Sciences, New York, USA, he joined the faculty of the University of Washington, Seattle, USA, where he is now professor emeritus of mathematics. He has written a number of research and expository articles on harmonic analysis and its applications, and he is the author of eleven textbooks and research monographs.

About the Series

Textbooks in Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Functional Analysis