2nd Edition

A Course in Abstract Harmonic Analysis

By Gerald B. Folland Copyright 2015
    320 Pages 3 B/W Illustrations
    by Chapman & Hall

    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.

    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


    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.

    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