Abstract

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.

中文翻译：

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一系列指数单项式之和的存在域

摘要

在本文中，考虑了一系列指数单项式。我们研究级数和的奇异点在其收敛域边界上的分布问题。我们研究的条件是，对于具有选定收敛域的级数的任何系数序列，该级数之和的存在域与给定收敛域一致。我们考虑具有角密度（可测量）和零凝聚指数的指数序列。得到了与一系列指数单项式之和在其收敛域边界上的奇异点分布有关的各种判据。特别地，在指示序列的类别中，获得这样的准则：所选凸域的所有边界点对于具有给定收敛域的级数的任何和都是特殊的。该标准是使用指数序列和凸域（角密度和边界弧的长度）的简单几何特征来制定的。还表明，凝聚指数等于0的条件是必要的。