On summation of non-harmonic Fourier series

Abstract: Let a sequence $\Lambda\subset\mathbb{C}$ be such that the corresponding system of exponential functions $\mathcal{E}(\Lambda):={e^{i\lambda t}}_{\lambda\in\Lambda}$ is complete and minimal in $L^2(-\pi,\pi)$ and thus each function $f\in L^2(-\pi,\pi)$ corresponds to a non-harmonic Fourier series in $\mathcal{E}(\Lambda)$. We prove that if the generating function $G$ of $\Lambda$ satisfies Muckenhoupt $(A_2)$ condition on $\mathbb{R}$, then this series admits a linear summation method. Recent results show that $(A_2)$ condition cannot be omitted.