Harmonic Analysis on Rotation Groups: Abel Summability

Abstract

The fundamental theorem of harmonic analysis on compact groups is just the Peter-Weyl theorem. It claims that every continuous function on a compact group can be approximated as closely as desired by linear combinations of irreducible representations of the group. Professor Hua modified this theorem in Ref. 1. He defined Fourier series for continuous functions on the unitary group U(n) and proved that any continuous function on the unitary group can be obtained through Fourier series via Abel summability.