@article{Favorov_2024, title={Almost periodic distributions and crystalline measures}, volume={61}, url={http://www.matstud.org.ua/ojs/index.php/matstud/article/view/467}, DOI={10.30970/ms.61.1.97-108}, abstractNote={<p>We study temperate distributions and measures with discrete support in Euclidean space and their Fourier transforms<br>with special attention to almost periodic distributions. In particular, we prove that if distances between points of the support of a measure do not quickly approach 0 at infinity, then this measure is a Fourier quasicrystal (Theorem 1).</p> <p>We also introduce a new class of almost periodicity of distributions,<br>close to the previous one, and study its properties.<br>Actually, we introduce the concept of s-almost periodicity of temperate distributions. We establish the conditions for a measure $\mu$ to be s-almost periodic (Theorem 2), a connection between s-almost periodicity<br>and usual almost periodicity of distributions (Theorem 3). We also prove that the Fourier transform of an almost periodic distribution with locally finite support is a measure (Theorem 4),<br>and prove a necessary and sufficient condition on a locally finite set $E$ for each measure with support on $E$ to have s-almost periodic Fourier transform (Theorem 5).</p>}, number={1}, journal={Matematychni Studii}, author={Favorov, S. Yu.}, year={2024}, month={Mar.}, pages={97-108} }