Absolute convergence of Fourier series in eigenfunctions of an elliptic operator

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 $B_{2,\theta}^{N/2}$. 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 $B_{2,\theta}^{N/2}$ whose Fourier series is absolutely divergent at any preassigned point.