Abstract:
Let G be a convex domain with support function h(−φ), and let {λk} be distinct complex numbers. In this paper the author determines when the system
{eλkz} is absolutely representing in the space A(G) of functions analytic in G, with the topology of uniform convergence on compact sets. In particular he proves the
Theorem. {\it Let L(λ) be an exponential function with indicator h(φ) and simple zeros {λn}∞n=1. For the system {eλkz}∞k=1 to be absolutely representing in A(G) it is necessary and sufficient that either of the following two conditions hold}:
1) {\it The system {eλkz}∞k=1 has a nontrivial expansion of zero in A(G), i.e. ∑∞n=1bneλnz=0 for every z∈G}.
\smallskip
2) L(λ) is a function of completely regular growth and there exists a function C(λ) of class [1,0] such that ¯limn→∞[1|λn|ln|C(λn)L′(λn)|+h(argλn)]⩽0.
\Bibitem{Kor80}
\by Yu.~F.~Korobeinik
\paper Interpolation problems, nontrivial expansions of zero, and representing systems
\jour Math. USSR-Izv.
\yr 1981
\vol 17
\issue 2
\pages 299--337
\mathnet{http://mi.mathnet.ru/eng/im1951}
\crossref{https://doi.org/10.1070/IM1981v017n02ABEH001355}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=595258}
\zmath{https://zbmath.org/?q=an:0471.30003|0445.30004}
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Linking options:
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This publication is cited in the following 25 articles:
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Some properties of absolutely representing systems of exponential functions and partial fractions in spaces of holomorphic functions with given boundary smoothness
Yu. F. Korobeinik, “Nekotorye voprosy teorii lineinykh topologicheskikh prostranstv (polnota, netrivialnye razlozheniya nulya i porozhdayuschie ikh elementy)”, Vladikavk. matem. zhurn., 12:3 (2010), 47–55
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A. S. Krivosheev, “A fundamental principle for invariant subspaces in convex domains”, Izv. Math., 68:2 (2004), 291–353
Yu. F. Korobeinik, “O lineinykh sistemakh uravnenii v operatorakh obobschennoi svertki”, Vladikavk. matem. zhurn., 6:2 (2004), 39–49
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
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
Korobeinik Y.F., “The Fourier method in the Cauchy problem and absolutely representing systems of exponentials. I”, Differential Equations, 35:12 (1999), 1693–1701
A. V. Abanin, “Nontrivial expansions of zero and absolutely representing systems”, Math. Notes, 57:4 (1995), 335–344
Yu. F. Korobeinik, “Nontrivial expansions of zero in absolutely representing systems. Application to convolution operators”, Math. USSR-Sb., 73:1 (1992), 49–66
Yu. F. Korobeinik, “Description of the general form of nontrivial expansions of zero in exponentials. Applications”, Math. USSR-Izv., 39:2 (1992), 1013–1032