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Mat. Zametki, 1971, Volume 10, Issue 1, Pages 57–62 (Mi mz7067)  

Theorem concerning analytic continuation

A. M. Lukatskii

M. V. Lomonosov Moscow State University

Abstract: A. I. Markushevich obtained the following representation of a function in its holomorphicity star with a sequence $\{m_\nu\}$, for which $m_{\nu+1}/m_\nu\to\infty$:
$$f(z)=\lim\limits_{\nu\to\infty}\{\sum_0^{m_{2\nu}}\theta_k\frac{f^{(k)}(z_0)}{k!}(z-z_0)^k+\sum_0^{m_{2\nu-1}}(1-\theta_k)\frac{f^{(k)}(z_0)}{k!}(z-z_0)^k\}$$
. Here it is proved that this condition is necessary; more precisely, $\overline{\lim\limits_{\nu\to\infty}}\frac{m_{\nu+1}}{m_\nu}=\infty$ . This result is derived from certain properties of over-convergent power series.

Full text: PDF file (331 kB)

English version:
Mathematical Notes, 1971, 10:1, 459–462

Bibliographic databases:

UDC: 517.5
Received: 07.04.1970

Citation: A. M. Lukatskii, “Theorem concerning analytic continuation”, Mat. Zametki, 10:1 (1971), 57–62; Math. Notes, 10:1 (1971), 459–462

Citation in format AMSBIB
\Bibitem{Luk71}
\by A.~M.~Lukatskii
\paper Theorem concerning analytic continuation
\jour Mat. Zametki
\yr 1971
\vol 10
\issue 1
\pages 57--62
\mathnet{http://mi.mathnet.ru/mz7067}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=286987}
\zmath{https://zbmath.org/?q=an:0215.12401}
\transl
\jour Math. Notes
\yr 1971
\vol 10
\issue 1
\pages 459--462
\crossref{https://doi.org/10.1007/BF01747070}


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