×

zbMATH — the first resource for mathematics

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, \(\nu=0,\cdots,k\).
MSC:
30B40 Analytic continuation of functions of one complex variable
11J71 Distribution modulo one
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] L. J. Mordell,Proc. Amer. Math. Soc.,12, 522–526 (1961). · doi:10.1090/S0002-9939-1961-0125947-0
[2] L. J. Mordell,Notices Amer. Math. Soc.,11, 312 (1964).
[3] A. I. Pavlov,Mat. Zametki [Math. Notes],55, No. 2, 102–108 (1994).
[4] L. Bieberbach,Analytische Fortzetzung, Springer, Berlin-Göttingen-Heidelberg (1955).
[5] L. Kuipers and H. Niederreiter,Uniform Distribution of Sequences, Interscience, New York (1974). · Zbl 0281.10001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.