On summation of non-harmonic Fourier series

Let a sequence $Λ\subset\mathbb{C}$ be such that the corresponding system of exponential functions $\mathcal{E}(Λ):=\{e^{iλt}\}_{λ\inΛ}$ is complete and minimal in $L^2(-π,π)$ and thus each function $f\in L^2(-π,π)$ corresponds to a non-harmonic Fourier series in $\mathcal{E}(Λ)$. We prove that if the generating function $G$ of $Λ$ 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.

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