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Localization of singularities of power series on the boundary of the disk of convergence. (Russian. English summary) Zbl 0626.30002

In the paper the authors investigate questions concerning the famous theorem of Fabry about singularities of power series of the form \[ (1)\quad \sum^{\infty}_{n=0}f_ nz^ n,\quad \overline{\lim}_{n\to \infty}| f_ n|^{1/n}=1. \] One of the main results is Theorem 1. Let two sequences \(\{n_ k\}_ 1^{\infty}\), \(n_ k\uparrow +\infty\), \(n_ k\in Z_+\), and \(\{\beta_ k\}_ 1^{\infty}\), \(\beta_ k\in {\mathbb{R}}\), so that \[ \overline{\lim}_{k\to \infty}| Re(f_{n_ k}e^{i\beta_ k})|^{1/n_ k}=1, \] where \(f_ n\) are the coefficients of the series (1).
Then the series (1) has a singular point \(e^{i\omega}\), where \[ | \omega | \leq \pi \min \{\overline{\lim}_{\tau \to 1- 0}\overline{\lim}_{k\to \infty}(W_ k(\tau))/(| 1-\tau | n_ k),\quad \overline{\lim}_{\tau \to 1+0}\overline{\lim}_{k\to \infty}(W_ k(\tau))/(| 1-\tau | n_ k)\} \] and \(W_ k(\tau)\) is the number of sign changes in the sequence \(\{Re(f_ ne^{i\beta_ k})\}\) for \(n\in [\tau n_ k,n_ k]\), \(\tau\geq 0.\)
From the main results the authors receive some corollaries which generalize Fabry’s theorems about gaps and ratios for power series.
Reviewer: P.Z.Agranovich

MSC:

30B10 Power series (including lacunary series) in one complex variable
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