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.

arXiv comment16 pages, We modified subsections 4.3, 4.4 and added explanations in subsection 4.5