On Summation of Nonharmonic Fourier Series

Abstract

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