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<p>We consider algebras of entire functions of exponential type that are defined by additional growth restrictionsalong the&#xa0;real line: the Bernstein algebra, the Schwartz algebra, and the Beurling–Björck algebra.The first of these algebras consists of entire functions of exponential type that are bounded on the real line.The Schwartz algebra consists of entire functions of exponential type whose growth along the real axis does not exceed polynomial growth, and the Beurling–Björck algebra is defined as the algebra of entire functions of&#xa0;exponential type whose growth along the real axis is bounded by a special weight function.We prove a criterion for an entire function to be a divisor of the Bernstein algebra in terms of the so-called “slow decrease.”Similar criteria are well known for the Schwartz and Beurling–Björck algebras, which are important in applications.We also describe a&#xa0;connection between the set of divisors of the Bernstein algebra and the class of sine-type functions.In&#xa0;the second part of this paper, some conditions are given for the shift of an integer sequence, under which the perturbed sequence is the zero set of a&#xa0;divisor in&#xa0;each of the algebras under consideration.The&#xa0;corresponding criterion for the Beurling–Björck algebra is obtained.It is emphasized that, in general, these conditions on admissible shifts of an&#xa0;integer sequence for all three algebras depend equally on the weight that bounds the growth of functions along the real&#xa0;line.</p>

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Divisors in Some Weighted Algebras of Entire Functions

  • 期刊论文

  • 发表日期：

    2025年9月30日

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  • N. F. Abuzyarova,
  • D. V. Semenova

摘要

We consider algebras of entire functions of exponential type that are defined by additional growth restrictionsalong the real line: the Bernstein algebra, the Schwartz algebra, and the Beurling–Björck algebra.The first of these algebras consists of entire functions of exponential type that are bounded on the real line.The Schwartz algebra consists of entire functions of exponential type whose growth along the real axis does not exceed polynomial growth, and the Beurling–Björck algebra is defined as the algebra of entire functions of exponential type whose growth along the real axis is bounded by a special weight function.We prove a criterion for an entire function to be a divisor of the Bernstein algebra in terms of the so-called “slow decrease.”Similar criteria are well known for the Schwartz and Beurling–Björck algebras, which are important in applications.We also describe a connection between the set of divisors of the Bernstein algebra and the class of sine-type functions.In the second part of this paper, some conditions are given for the shift of an integer sequence, under which the perturbed sequence is the zero set of a divisor in each of the algebras under consideration.The corresponding criterion for the Beurling–Björck algebra is obtained.It is emphasized that, in general, these conditions on admissible shifts of an integer sequence for all three algebras depend equally on the weight that bounds the growth of functions along the real line.