Equiconvergence and equisummability of nonharmonic Fourier expansions with ordinary trigonometric series

Given $f\in L(-\pi,\pi)$, we consider its nonharmonic Fourier series $f(x)\sim\sum c_ne^{i\lambda}n^x$, where $\lambda_n$ are the roots of the entire function $L(z)=\int_{-\pi}^\pi e^{izt}\,d\sigma(T)$. We show that this series is equiconvergent, uniformly inside $(-\pi,\pi)$, and equisummable with the Fourier series of $f$ with respect to the trigonometric system if $\sigma'(t)=k(t)(\pi-|t|)^{-\alpha}$, $\alpha\in(0,1)$, $\operatorname{var}k<\infty$, $k(\pi-0)\ne0$, $k(-\pi+0)\ne0$.