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Izv. Akad. Nauk SSSR Ser. Mat., 1988, Volume 52, Issue 5, Pages 991–1004 (Mi izv1214)  

This article is cited in 7 scientific papers (total in 7 papers)

On expansion of analytic functions in exponential series

S. N. Melikhov


Abstract: Let $G$ be an arbitrary convex domain in the $p$-dimensional ($p\in\mathbf N$) complex space $\mathbf C^p$, and $H(G)$ the space of single-valued analytic functions on $G$, endowed with the topology $\tau_G$ of uniform convergence on compact subsets of $G$. In this paper the following assertion is obtained (as a corollary to a more general result proved here) for a bounded domain $G$: if a sequence $\{E_n\}_{n\in\mathbf N}$ of closed subspaces of $H(G)$ that are invariant under each partial differentiation $\frac{\partial}{\partial z_k}$ ($k=1,…,p$) has the property that every function locally analytic on $\overline G$ can be represented as a series
\begin{equation} \sum_{n=1}^\infty x_n(z),\qquad x_n(z)\in E_n,\quad\forall n\in\mathbf N, \end{equation}
convergent (absolutely convergent) in the topology $\tau_G$, then any function in $H(G)$ can be expanded in a series (1) convergent (absolutely convergent) in $\tau_G$.
Bibliography: 21 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1989, 33:2, 317–329

Bibliographic databases:

UDC: 517.9
MSC: Primary 32A05, 32A30, 30B50; Secondary 46E10, 46A05, 46A12
Received: 22.05.1986
Revised: 07.05.1987

Citation: S. N. Melikhov, “On expansion of analytic functions in exponential series”, Izv. Akad. Nauk SSSR Ser. Mat., 52:5 (1988), 991–1004; Math. USSR-Izv., 33:2 (1989), 317–329

Citation in format AMSBIB
\Bibitem{Mel88}
\by S.~N.~Melikhov
\paper On~expansion of analytic functions in exponential series
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1988
\vol 52
\issue 5
\pages 991--1004
\mathnet{http://mi.mathnet.ru/izv1214}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=972092}
\zmath{https://zbmath.org/?q=an:0679.32003|0661.32001}
\transl
\jour Math. USSR-Izv.
\yr 1989
\vol 33
\issue 2
\pages 317--329
\crossref{https://doi.org/10.1070/IM1989v033n02ABEH000829}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. N. Melikhov, “Extension of entire functions of completely regular growth and right inverse to the operator of representation of analytic functions by quasipolynomial series”, Sb. Math., 191:7 (2000), 1049–1073  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. S. N. Melikhov, E. V. Teknechyan, “On the expansion of analytic functions in series in successive derivatives”, Russian Math. (Iz. VUZ), 47:2 (2003), 74–78  mathnet  mathscinet  zmath  elib
    3. Sergej N. Melikhov, “(DFS)-spaces of holomorphic functions invariant under differentiation”, Journal of Mathematical Analysis and Applications, 297:2 (2004), 577  crossref
    4. S. N. Melichow, “Über absolut repräsentierende Systeme aus Quasipolynomen in Räumen analytischer Funktionen”, Math Nachr, 158:1 (2006), 299  crossref
    5. V. B. Sherstyukov, “Nontrivial expansions of zero and representation of analytic functions by series of simple fractions”, Siberian Math. J., 48:2 (2007), 369–381  mathnet  crossref  mathscinet  zmath  isi  elib
    6. S. N. Melikhov, “Koeffitsienty ryadov eksponent dlya analiticheskikh funktsii i operator Pomme”, Kompleksnyi analiz. Tselye funktsii i ikh primeneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 161, VINITI RAN, M., 2019, 65–103  mathnet
    7. V. B. Sherstyukov, “Asimptoticheskie svoistva tselykh funktsii s zadannym zakonom raspredeleniya kornei”, Kompleksnyi analiz. Tselye funktsii i ikh primeneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 161, VINITI RAN, M., 2019, 104–129  mathnet
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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