On the summability and convergence of non-harmonic Fourier series

We consider systems of exponentials that are orthogonal to measures $d\sigma$ of a special form on $(-a,a)$. Under certain conditions on the summation method, these systems form summation bases $L^p(-a,a)$ and in $C_0$ (the subspace of $C[-a,a]$ orthogonal to $d\sigma$). With respect to these systems, Lipschitzian functions in $C_0$ are expanded into non-harmonic Fourier series that converge uniformly on $[-a,a]$.