Some properties of Fourier quasicrystals and measures on a strip

Abstract

In our paper we extend some results of the theory of Fourier quasicrystals on the real line to a horizontal strip of finite width. For measures in a strip we use a natural generalization of the usual Fourier transform for measures on the line. We consider positive or translation bounded measures $\mu$ on a strip whose Fourier transform is a pure point measure $\hat\mu=\sum_{\gamma\in\Gamma}b_\gamma\delta_\gamma$ (as usual, $\delta_\gamma$ is the unit mass at the point $\gamma$). We prove that the measure $\nu=\sum_{\gamma\in\Gamma}|b_\gamma|^2\delta_\gamma$ has the exponential growth. Moreover, if for some $\eta>0$ the points of $\Gamma$ in every interval of length $\eta$ are linearly independent over integers, then the measure $\hat\mu$ also has the exponential growth.