Harmonic analysis on the Heisenberg group

Author(s)

Bibliographic Information

Harmonic analysis on the Heisenberg group

Sundaram Thangavelu

（Progress in mathematics, v. 159）

Birkhäuser, c1998

  • : us
  • : gw

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Note

Includes bibliographical references (p. [185]-191) and index

Description and Table of Contents

Volume

: us ISBN 9780817640507

Description

The Heisenberg group plays an important role in several branches of mathematics, such as representation theory, partial differential equations, number theory, several complex variables and quantum mechanics. This monograph deals with various aspects of harmonic analysis on the Heisenberg group, which is the most commutative among the non-commutative Lie groups, and hence gives the greatest opportunity for generalizing the remarkable results of Euclidean harmonic analysis. The aim of this text is to demonstrate how the standard results of abelian harmonic analysis take shape in the non-abelian setup of the Heisenberg group. Thangavelu's exposition is clear and well developed, and leads to several problems worthy of further consideration. Any reader who is interested in pursuing research on the Heisenberg group will find this unique and self-contained text invaluable.

Table of Contents

1 The Group Fourier Transform.- 1.1 The Heisenberg group.- 1.2 The Schroedinger representations.- 1.3 The Fourier and Weyl transforms.- 1.4 Hermite and special Hermite functions.- 1.5 Paley-Wiener theorems for the Fourier transform.- 1.6 An uncertainty principle on the Heisenberg group.- 1.7 Notes and references.- 2 Analysis of the Sublaplacian.- 2.1 Spectral theory of the sublaplacian.- 2.2 Spectral decomposition for Lp functions.- 2.3 Restriction theorems for the spectral projections.- 2.4 A Paley-Wiener theorem for the spectral projections.- 2.5 Bochner-Riesz means for the sublaplacian.- 2.6 A multiplier theorem for the Fourier transform.- 2.7 Notes and references.- 3 Group Algebras and Applications.- 3.1 The Heisenberg motion group.- 3.2 Gelfand pairs, spherical functions and group algebras.- 3.3 An algebra of radial measures.- 3.4 Analogues of Wiener-Tauberian theorem.- 3.5 Spherical means on the Heisenberg group.- 3.6 A maximal theorem for spherical means.- 3.7 Notes and references.- 4 The Reduced Heisenberg Group.- 4.1 The reduced Heisenberg group.- 4.2 A Wiener-Tauberian theorem for Lp functions.- 4.3 A maximal theorem for spherical means.- 4.4 Mean periodic functions on phase space.- 4.5 Notes and references.
Volume

: gw ISBN 9783764340506

Description

The Heisenberg group has been implicitly present in much of mathematics and physics for a long time. Thangavelu's work is developed within the framework of harmonic analysis (Fourier transforms, convolution algebra, and related ideas) and gives a survey of the field. Additionally the author discusses in detail the representation theory of the group and its relationship to the theory of classical special functions. Among the topics covered are the Plancherel and Paley-Wiener theorems, spectral theory of the sublaplacian, Wiener-Tauberian theorems, Bocher-Riesz means and multipliers for the Fourier transform. The exposition is systematic, and leads to several problems and conjectures for further consideration. Any reader who is interested in pursuing research on the Heisenberg group should find this text of use.

Table of Contents

  • The group Fourier transform
  • analysis of the sublaplacian
  • group algebras and applications
  • the reduced Heisenberg group.

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