Analytic continuation of the Fourier series on connected compact Lie groups

AbstractFor a connected compact real Lie group K with complexification G, we study the class of C∞ functions on K having an analytic continuation into a neighborhood of K in G. The main result describes such a function in terms of its Fourier coefficients, showing that they have to decrease exponentially. The region of analyticity naturally arising from the analytic continuation of the Peter-Weyl expansion is shown to be a domain of holomorphy. These results yield as an application the Gegenbauer polynomial expansion of an analytic function