Singular points of the sum of a series of exponential monomials on the boundary of the convergence domain

Krivosheyeva, O.

doi:10.1090/S1061-0022-2012-01199-4

Singular points of the sum of a series of exponential monomials on the boundary of the convergence domain
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by O. A. Krivosheyeva
Translated by: S. Kislyakov

St. Petersburg Math. J. 23 (2012), 321-350

DOI: https://doi.org/10.1090/S1061-0022-2012-01199-4

Published electronically: January 24, 2012

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Abstract:

Singular points for the sum of a series of exponential monomials are studied. The main statement contains results of Hadamard, Fabry, V. Bernstein, Polya, Carlson and Landau as particular cases. Moreover, a special function is constructed that has no singular points on the boundary of the convergence domain of its series. This function generalizes a certain special function in the theory of Dirichlet series to the case of series of exponential monomials. The existence of this special function shows the necessity of a condition in the main theorem; in V. Bernstein’s theorem, a similar role is played by the requirement that the condensation index should be equal to zero.

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