Fourier transforms

Abstract

This chapter presents basic facts about Fourier transforms on both locally compact Abelian and arbitrary compact groups, many of which are needed for the detailed analysis in Chapters Nine, Ten, and Eleven. The Fourier transform as defined in (23.9) is a complex-valued function on the character group of a locally compact Abelian group. For a compact non-Abelian group, the Fourier transform is an operator-valued function defined on the dual object (28.34). Parts of the theory can be described simultaneously for compact and for locally compact Abelian groups, and we will do this wherever possible.