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

This article is cited in 22 scientific papers (total in 23 papers)

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
Received: 14.11.1968

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
\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
\jour Russian Math. Surveys
\yr 1969
\vol 24
\issue 2
\pages 101--178

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    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  mathnet  crossref  mathscinet  zmath
    2. A. F. Leont'ev, “On representation by Dirichlet series of functions analytic in a halfplane”, Math. USSR-Sb., 14:4 (1971), 565–581  mathnet  crossref  mathscinet  zmath
    3. A. M. Sedletskii, “On biorthogonal expansions in exponential functions”, Math. USSR-Izv., 6:3 (1972), 579–586  mathnet  crossref  mathscinet  zmath
    4. A. F. Leont'ev, “On conditions of expandibility of analytic functions in Dirichlet series”, Math. USSR-Izv., 6:6 (1972), 1265–1277  mathnet  crossref  mathscinet  zmath
    5. A. F. Leont'ev, “On representing entire functions of several variables by Dirichlet series”, Math. USSR-Sb., 18:4 (1972), 589–602  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    7. M. M. Dzhrbashyan, “Uniqueness theorems for analytic functions asymptotically representable by Dirichlet–Taylor series”, Math. USSR-Sb., 20:4 (1973), 603–649  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    9. V. K. Dzyadyk, “On convergence conditions for Dirichlet series on closed polygons”, Math. USSR-Sb., 24:4 (1974), 463–481  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    11. Yu. I. Mel'nik, “On the representation of regular functions by Dirichlet series in a closed disk”, Math. USSR-Sb., 26:4 (1975), 449–457  mathnet  crossref  mathscinet  zmath
    12. I. S. Galimov, “On the completeness of the exponential system in nonconvex domains”, Math. USSR-Sb., 27:1 (1975), 39–50  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    15. V. A. Tkachenko, “Spectral decompositions in spaces of analytic functionals”, Math. USSR-Izv., 14:3 (1980), 597–651  mathnet  crossref  mathscinet  zmath  isi
    16. Carlos A. Berenstein, B.A. Taylor, “A new look at interpolation theory for entire functions of one variable”, Advances in Mathematics, 33:2 (1979), 109  crossref
    17. V. A. Tkachenko, “Spectral theory in spaces of analytic functionals for operators generated by multiplication by the independent variable”, Math. USSR-Sb., 40:3 (1981), 387–427  mathnet  crossref  mathscinet  zmath  isi
    18. Yu. F. Korobeinik, “Interpolation problems, nontrivial expansions of zero, and representing systems”, Math. USSR-Izv., 17:2 (1981), 299–337  mathnet  crossref  mathscinet  zmath  isi
    19. Yu. F. Korobeinik, “Representing systems”, Russian Math. Surveys, 36:1 (1981), 75–137  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    21. Carlos A. Berenstein, Bao Qin Li, “Interpolating varieties for spaces of meromorphic functions”, J Geom Anal, 5:1 (1995), 1  crossref
    22. Bao Qin Li, Enrique Villamor, “Interpolating multiplicity varieties in C n”, J Geom Anal, 11:1 (2001), 91  crossref
    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  mathnet  crossref  elib
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