The Peter–Weyl theorem & harmonic analysis on S^n

Abstract

For finite groups, the Artin–Wedderburn theorem gives a precise decomposition of the algebra of all C-valued functions into matrix algebras. Specialized to the case of cyclic groups, this produces the classical discrete Fourier transform. In this paper, we endeavor to extend these techniques to compact topological groups, proving similar harmonic decompositions on S¹, S², and S³.