On summation of non-harmonic Fourier series

Let a sequence $\mathrm{\Lambda }\subset \mathbb{C}$ be such that the corresponding system of exponential functions $$\mathcal{E}$(\Lambda):={e^{i\lambda t}}_{\lambda\in\Lambda}$iscompleteandminimalin$L^2(-\pi,\pi)$ and thus each function $f\in L^2(-\pi ,\pi )$ corresponds to a non-harmonic Fourier series in $\mathcal{E}(\mathrm{\Lambda })$. We prove that if the generating function $G$ of $\mathrm{\Lambda }$ 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.