Some properties of Fourier quasicrystals and measures on a strip

Abstract

We extend certain results of the theory of Fourier quasicrystals on the real line to the case of 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.

9 pages, 18 references