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

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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.

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Funding

The work of the second author was financially supported by the competition “Young Mathematics of Russia.”

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Authors and Affiliations

  1. Institute of Mathematics with Computing Centre—Subdivision of the Ufa Federal Research Centre of Russian Academy of Sciences, Ufa, 450008, Russia

    A. S. Krivosheev

  2. Ufa University of Science and Technology, Ufa, ums@uust.ru, Russia

    O. A. Krivosheeva

Authors
  1. A. S. Krivosheev
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  2. O. A. Krivosheeva
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Corresponding author

Correspondence to O. A. Krivosheeva.

Additional information

Translated from Matematicheskie Zametki, 2023, Vol. 114, pp. 563–578 https://doi.org/10.4213/mzm13794.

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Krivosheev, A.S., Krivosheeva, O.A. Domain of Existence of the Sum of a Series of Exponential Monomials. Math Notes 114, 508–521 (2023). https://doi.org/10.1134/S0001434623090213

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  • DOI: https://doi.org/10.1134/S0001434623090213

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