Sugiura M. Unitary Representations and Harmonic Analysis: An Introduction

North Holland, 1990. — 469 p.The principal aim of this book is to give an introduction to harmonic analysis and the theory of unitary representations of Lie groups. The second edition has been brought up to date with a number of textual changes in each of the five chapters, a new appendix on Fatou's theorem has been added in connection with the limits of discrete series, and the bibliography has been tripled in length.Preface to the Second Edition
Preface
Conventions and Notations
Fourier series and the torus group T
Introduction
Fundamental definitions
Unitary representations of compact groups
Fourier series of square integrable functions
Fourier series of smooth functions and distributions
Representations of SU(2) and SO(3)
Construction of irreducible representations of SU(2)
Characters of compact groups
Haar measures on SU(2)
Enumeration of irreducible representations
Lie algebras and their representations
Fourier series on SU(2)
Representations of SO(3) and spherical harmonics
Fourier series on compact Lie groups
The Fourier transform and unitary representations of Rn
Rapidly decreasing functions
The Plancherel theorem and the decomposition of the regular representation
Positive definite functions and Stone’s theorem
The Paley-Wiener theorem
Tempered distributions and their Fourier transforms
The Euclidean motion group
Construction of irreducible representations
Classification of irreducible unitary representations
Fourier transforms of rapidly decreasing functions
The Plancherel theorem
Determination of g(G) and D(G)
Unitary representation of SL(2, R)
The Iwasawa decomposition
Irreducible unitary representations
Irreducible unitary representations
Irreducible unitary representations
K-finite vectors
Classification of irreducible unitary representations
The characters
Inversion formula
Harmonic analysis of zonal functions
Irreducible unitary representations of SL (2, R)
Appendix
Notes
Bibliography
Index