Fourier Transforms and Duality

Abstract

One of the directions of the theory of representations of groups, which has been given the name of harmonic analysis, consists in the study of function spaces on groups and homogeneous spaces in terms of the representations that appear in them. Here a great rôle is played by the generalized Fourier transform. As is known, the classical Fourier transform on the real line carries translations into multiplication by functions and thus gives the spectral decomposition of an arbitrary operator that commutes with translations (for example, a differential operator with constant coefficients). The generalized Fourier transform is designed to play exactly this rôle for an arbitrary group. This transform is defined as follows. Let G be a topological group and ĜG the set of equivalence classes of irreducible unitary representations of G. For each class λ∈Ĝ, we choose a representation T_λ belonging to this class and acting in a Hilbert space H_λ.