Unitary Representations and Harmonic Analysis

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. Determinations of Ĝ(G) and D(G). Unitary Representation of SL(2, R). The Iwasawa Decomposition. Irreducible Unitary Representations: I. Principal Continuous Series. II. Principal Discrete Series. III. The Limit of Discrete Series. IV. Complementary Series. K-Finite Vectors. Classification of Irreducible Unitary Representations. The Characters. Inversion Formula. Harmonic Analysis of Zonal Functions. Irreducible Unitary Representations of SL (2, R): I. Discrete Series. II. Complementary Series. III. Principal Continuous Series. Appendix. Bibliography. Index.