Domain of Existence of the Sum of a Series of Exponential Monomials

• Published in: Mathematical Notes 2023-10, Vol.114 (3-4), p.508-521
• Main Authors: Krivosheev, A. S., Krivosheeva, O. A.
• Format: Article
• Language: English
• Subjects: 14/34; 639/766/189; 639/766/530; 639/766/747; Convergence; Criteria; Density; Exponents; Mathematics; Mathematics and Statistics; Sequences; Series (mathematics)
• ISSN: 0001-4346; 1067-9073; 1573-8876
• DOI: 10.1134/S0001434623090213

In the paper, series of exponential monomials are considered. We study the problem of the distribution of singular points of the sum of a series on the boundary of its domain of convergence. We study the conditions under which, for any sequence of coefficients of the series with a chosen domain of convergence, the domain of existence of the sum of this series coincides with the given domain of convergence. We consider sequences of exponents having an angular density (measurable) and the zero condensation index. Various criteria related to the distribution of singular points of the sum of a series of exponential monomials on the boundary of its convergence domain are obtained. In particular, in the class of the indicated sequences, a criterion is obtained that all boundary points of a chosen convex domain are special for any sum of a series with a given domain of convergence. The criteria are formulated using simple geometric characteristics of the sequence of exponents and a convex domain (the angular density and the length of the boundary arc). It is also shown that the condition that the condensation index is equal to zero is essential.