Mean convergence of Fourier series on compact Lie groups

Abstract

The main result is an L p {L^p} mean convergence theorem for the partial sums of the Fourier series of a class function on a compact semi-simple Lie group. A central element in the proof is a Lie group-Lie algebra analog of the theorems in classical Fourier analysis that allow one to pass back and forth between multiplier operators for Fourier series in several variables and multiplier operators for the Fourier transform in Euclidean space. To obtain the L p {L^p} mean convergence theorem, the theory of the Hilbert transform with weight function is needed.