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Mat. Zametki, 1999, Volume 66, Issue 4, Pages 540–550 (Mi mz3987)  

Entire functions, analytic continuation, and the fractional parts of a linear function

A. I. Pavlov

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: The main result of the paper is as follows.
Theorem. Suppose that $G(z)$ is an entire function satisfying the following conditions:
1) the Taylor coefficients of the function $G(z)$ are nonnegative;
2) for some fixed $C>0$ and $A>0$ and for $|z|>R_0$, the following inequality holds:
$$ |G(z)|<\exp(C\frac{|z|}{\ln^A|z|}). $$
{\it Further, suppose that for some fixed $\alpha>0$ the deviation $D_N$ of the sequence $x_n=\{\alpha n\}$, $n=1,2,…$, as $N\to\infty$ has the estimate $D_N=O(\ln^BN/N)$. Then if the function $G(z)$ is not an identical constant and the inequality $B+1<A$ holds, then the power series $\sum_{n=0}^\infty G([\alpha n])z^n$ converging in the disk $|z|<1$ cannot be analytically continued to the region $|z|>1$ across any arc of the circle $|z|=1$.}

DOI: https://doi.org/10.4213/mzm3987

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English version:
Mathematical Notes, 1999, 66:4, 442–450

Bibliographic databases:

UDC: 517.5
Received: 03.12.1998

Citation: A. I. Pavlov, “Entire functions, analytic continuation, and the fractional parts of a linear function”, Mat. Zametki, 66:4 (1999), 540–550; Math. Notes, 66:4 (1999), 442–450

Citation in format AMSBIB
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\by A.~I.~Pavlov
\paper Entire functions, analytic continuation, and the fractional parts of a~linear function
\jour Mat. Zametki
\yr 1999
\vol 66
\issue 4
\pages 540--550
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\crossref{https://doi.org/10.4213/mzm3987}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1747082}
\elib{http://elibrary.ru/item.asp?id=13306018}
\transl
\jour Math. Notes
\yr 1999
\vol 66
\issue 4
\pages 442--450
\crossref{https://doi.org/10.1007/BF02679094}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000086188000027}


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