Дата: 11.02.2025 г., 14:00 ч.

Място: Зала 403, ИМИ – БАН

Докладчик: Sergey Favorov, Department of Pure Mathematics, Kharkov National University, Ukraine

Доклад: Fourier quasicrystals and their generalizations, zeros of Dirichlet series, other almost periodic objects

Допълнителна информация: https://icms.bg/fourier-quasicrystals-and-their-generalizations-zeros-of-dirichlet-series-other-almost-periodic-objects-icms-seminar-talk-by-sergey-favorov/

Резюме. A complex measure \(\mu\) on a \(d\)-dimensional Euclidean space is a crystalline measure (CM) if it is the temperate distribution, its distributional Fourier transform \(\hat\mu\) is also a measure, and supports of \(\mu\) and \(\hat\mu\) are discrete (locally finite); \(\mu\) is a Fourier quasicrystal (FQ) if, in addition, \(|\mu|\) and \(|\hat\mu|\) are also temperate distributions. For example, if \(\mu_0\) is the sum of the unit masses at all points with integer coordinates, then by Poisson’s formula \(\hat\mu_0=\mu_0\). Hence, \(\mu_0\) is FQ.

We show a theorem of Lev-Olevskii on a sufficient condition for trivialization of FQ. Then we discuss a simple condition for CM to be FQ and present CM that is not FQ.

We recall the notion of an almost periodic function, introduce the notions of almost periodic measures, distributions, sets, and show their connections with CM. In paricular, we get various uniqueness theorems for FQ.

Finally, we show the description of FQ with unit masses as zeros of exponential polynomials due to Olevskii and Ulanovskii, and discuss some generalizations to zeros of Dirichlet series and to measures in a horizontal strip of finite width.