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Analytic noncontinuability of certain classes of power series. (English. Russian original) Zbl 0972.30002
Math. Notes 64, No. 3, 410-413 (1998); translation from Mat. Zametki 64, No. 3, 474-477 (1998).
The author studies general classes of lacunary power series. Fix \(\alpha>0\) and \((m_{s}) _{s=1} ^\infty \subset\mathbb{N}\). Let \(\beta_{s} \) denote the fractional part of \(\alpha m_{s}\). Assume that \(\lim_{N \to+ \infty} (1/N)\# \{s\in{\mathbf N}: m_{s}\leq N\}>0\) and the sequence \((\beta _{s}) _{{s}=1} ^\infty\) is uniformly distributed in \([0,1]\). Let \(a_{s}: = f(\beta _{s})\), \(s\in\mathbb{N}\), where \(f\) is a Riemann-integrable function on \([0,1]\), whose Fourier coefficients can vanish only for a finite number of indices. Then the unit disc is the domain of existence of the function \(F(z)= \sum_{{s}1} ^\infty a_{s} z^{m_{s}}\). The same remains true if \(a_{s}:= \sum_{\nu=0}^{k} m_{s}^\nu p_\nu (\beta_{s})\), \(s\in \mathbb{N}\), where \(p_\nu\) is a nonconstant polynomial, \(\nu=0,\cdots,k\).
30B40 Analytic continuation of functions of one complex variable
11J71 Distribution modulo one
Full Text: DOI
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