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 Uspekhi Mat. Nauk, 1969, Volume 24, Issue 2(146), Pages 97–164 (Mi umn5474)

Representation of functions by generalized Dirichlet series

A. F. Leont'ev

Abstract: This article is concerned with the representation of functions in domains of the complex plain by series in the systems $\{e^{\lambda_{v}z}\}$, $\{f(\lambda_{v}z)\}$, $\{A(z, \lambda_\nu)\}$.
In § 1 we construct systems biorthogonal to the systems $\{e^{\lambda_{v}z}\}$, $\{f(\lambda_{v}z)\}$, $\{A(z, \lambda_\nu)\}$ and find the asymptotic behaviour of functions of these systems.
In § 2 we determine in a natural way the coefficients of the series in the systems in question by means of the biorthogonal systems. We also find the asymptotic behaviour of the coefficients for large indices. We obtain formulae for the remainder, that is, the difference between the function and a partial sum of the corresponding series.
In § 3 we prove that a function whose coefficients in the series are all zero must itself be zero. This result makes it possible, in principle, to reconstruct a function from the coefficients of its series. In § 4 we give conditions under which the series, or a subsequence of its partial sums, converges to the corresponding function.

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English version:
Russian Mathematical Surveys, 1969, 24:2, 101–178

Bibliographic databases:

UDC: 517.5
MSC: 30B50, 42C10, 30D10

Citation: A. F. Leont'ev, “Representation of functions by generalized Dirichlet series”, Uspekhi Mat. Nauk, 24:2(146) (1969), 97–164; Russian Math. Surveys, 24:2 (1969), 101–178

Citation in format AMSBIB
\Bibitem{Leo69} \by A.~F.~Leont'ev \paper Representation of functions by generalized Dirichlet series \jour Uspekhi Mat. Nauk \yr 1969 \vol 24 \issue 2(146) \pages 97--164 \mathnet{http://mi.mathnet.ru/umn5474} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=255811} \zmath{https://zbmath.org/?q=an:0195.35802|0176.02401} \transl \jour Russian Math. Surveys \yr 1969 \vol 24 \issue 2 \pages 101--178 \crossref{https://doi.org/10.1070/RM1969v024n02ABEH001343} 

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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. A. M. Sedletskii, “On functions periodic in the mean”, Math. USSR-Izv., 4:6 (1970), 1406–1428
2. A. F. Leont'ev, “On representation by Dirichlet series of functions analytic in a halfplane”, Math. USSR-Sb., 14:4 (1971), 565–581
3. A. M. Sedletskii, “On biorthogonal expansions in exponential functions”, Math. USSR-Izv., 6:3 (1972), 579–586
4. A. F. Leont'ev, “On conditions of expandibility of analytic functions in Dirichlet series”, Math. USSR-Izv., 6:6 (1972), 1265–1277
5. A. F. Leont'ev, “On representing entire functions of several variables by Dirichlet series”, Math. USSR-Sb., 18:4 (1972), 589–602
6. A. F. Leont'ev, “On an estimate for a Dirichlet polynomial and some of its applications”, Math. USSR-Sb., 20:4 (1973), 575–586
7. M. M. Dzhrbashyan, “Uniqueness theorems for analytic functions asymptotically representable by Dirichlet–Taylor series”, Math. USSR-Sb., 20:4 (1973), 603–649
8. A. F. Leont'ev, “On the representation of analytic functions in a closed convex region by a Dirichlet series”, Math. USSR-Izv., 7:3 (1973), 573–588
9. V. K. Dzyadyk, “On convergence conditions for Dirichlet series on closed polygons”, Math. USSR-Sb., 24:4 (1974), 463–481
10. B. Ya. Levin, Yu. I. Lyubarskii, “Interpolation by means of special classes of entire functions and related expansions in series of exponentials”, Math. USSR-Izv., 9:3 (1975), 621–662
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12. I. S. Galimov, “On the completeness of the exponential system in nonconvex domains”, Math. USSR-Sb., 27:1 (1975), 39–50
13. A. F. Leont'ev, “On the representation of analytic functions by series of exponentials in a polycylindrical domain”, Math. USSR-Sb., 29:3 (1976), 327–344
14. V. S. Vladimirov, S. M. Nikol'skii, Yu. N. Frolov, “Aleksei Fedorovich Leont'ev (on his sixtieth birthday)”, Russian Math. Surveys, 32:3 (1977), 131–144
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19. Yu. F. Korobeinik, “Representing systems”, Russian Math. Surveys, 36:1 (1981), 75–137
20. A. M. Sedletskii, “Biorthogonal expansions of functions in series of exponents on intervals of the real axis”, Russian Math. Surveys, 37:5 (1982), 57–108
21. Carlos A. Berenstein, Bao Qin Li, “Interpolating varieties for spaces of meromorphic functions”, J Geom Anal, 5:1 (1995), 1
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23. E. A. Kuryanovich, “Predstavlenie resheniya uravneniya Fridmana obobschennym ryadom Dirikhle”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 2(31) (2013), 200–205
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