On divisors in some weighted algebras of entire functions

N. F. Abuzyarova ; D. V. Semenova

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 4, pp. 5-20

Résumé

We consider three weighted algebras of entire functions of one variable.They are Bernstein algebra, Schwartz algebra and Beurling-Björck algebra. The Bernstein algebra is formed by entire functions of exponential type that are bounded on the real line. The Schwartz algebra consists of all entire functions of exponential type whose growth along the real axis does not exceed the polynomial one. And the Beurling-Björck algebra is defined as an algebra of entire functions of exponential type whose growth along the real axis is bounded by special weight function. We prove the criterion for divisors of the Bernsteing algebra in term of so-called «slow decrease». Similar criteria for the Scwartz algebra and the Beurling-Björck algebra are well-known. We also explore relations between the set of divisors of the Bernstein algebra and sine-type functions. In the second part of the work, conditions are given for the shift of an integer sequence under which the perturbed sequence is the zero set of the divisor of each of the algebras under consideration. The corresponding criterion for the Berling-Bjork algebra is obtained. It is emphasized that, in general, the conditions defining admissible shifts have the same form of dependence on the weight function in all three algebras.