Absolute convergence of Fourier series in eigenfunctions of an elliptic operator

Abstract

In this article we investigate absolute convergence of Fourier series in eigenfunctions of an m-th order elliptic operator on functions in the Besov class B 2, N/2 . We show that in terms of Besov classes the theorem of Peetre on absolute convergence of series in eigenfunctions in the class B 2,1 N/2 is best possible. We construct a function in B 2, N/2 whose Fourier series is absolutely divergent at any preassigned point.