The converse of the Fabry-Pólya theorem on singularities of lacunary power series

Martirosian, Valeri A.

doi:10.1515/anly-2005-0203

Cited by 2 publications

(1 citation statement)

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“…For the general case, a rather hard construction of a series (1.1) is presented in the monograph of Koosis [7,IX B], showing the sharpness of Theorem 1 in the sense, that for any fixed arc of length less than 2Á Ã ðP þ Þ a series of the form (1.1) exists which has no singular points on that arc. Finally, in the recent paper [8] of Martirosian a complete converse of Theorem 1 has been obtained, by constructing a series (1.1), for which there is a closed arc on @D 1 of the length 2Á Ã ðP þ Þ, with the property that all interior points of this arc are regular points for (1.1).…”

Section: The Problemmentioning

confidence: 99%

On the localization of singularities of Lacunar power series

Arakelian¹,

Luh

Müller

2007

Complex Variables and Elliptic Equations

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The main result of this paper is Theorem 4 on the existence of singularities of Lacunary power series on prescribed open boundary arcs of the circle of convergence. The influence of the gaps on the length of these arcs is expressed in terms of newly introduced integral densities. Theorem 4 contains the known Fabry-Po´lya theorem on gaps, describing the closed arcs having singularities, and suggests its extensions, using additional information on the coefficients of the power series. An essential step in its proof is Theorem 2, providing necessary and sufficient conditions (in terms of the so called ''Coefficient Functions'') for the analytic continuation of power series across fixed open boundary arcs.

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Section: The Problemmentioning

confidence: 99%