Fourier series on compact Lie groups

Abstract

In [I], Gong Shang proves that the Fourier series of a sufficiently smooth function on a unitary group U(n) converges absolutely and uniformly. Using the theory of elliptic operators, we give a short proof of a more general assertion. Suppose m is a finite strictly positive measure on a compact Cmanifold M. Suppose L is a strictly elliptic operator of order k on M which is selfadjoint on the space L2(M) of square integrable functions with respect to m. Then it is well known that for a sufficiently large positive scalar X, (X +L)-l is defined and compact, so L has a complete set of eigenfunctions oi in L2(M), and any function f L2(M) can be expanded in a Fourier series f = EZoi+x with respect to these eigenvectors, the series converging in Hilbert space norm. We ask the following question: what smoothness conditions can we place onf to guarantee that the sum f= Z (f. q.)0j converges absolutely and uniformly? The following theorem appears to be well known, but we shall record the proof, which is quite simple. In what follows, Hk, will be the Sobolev space of functions whose weak derivatives of order <ks are square integrable.