Abstract:
We have earlier proved (RZhMat., 1966, 2B149, 11B94) a theorem on the representation of an arbitrary function analytic in a closed convex region ¯D by a Dirichlet series in the open region D. In this paper we prove that any function analytic in an open convex finite region D and continuous in ¯D can be represented by a Dirichlet series with coefficients which can be computed by means of specific already-known formulas.
We also prove that if the convex region D is bounded by a regular analytic curve, then any function analytic in D can be expanded in a Dirichlet series in D. These two theorems are based on the following theorem from the theory of entire functions.
Let D be a finite open region, K(θ) the support function of D, h(θ)=H(−θ), and φ(r) a function satisfying the conditions
0<φ(r)↑∞,limr→∞lnφ(r)r=0.
Then there exists an entire function L(λ) of exponential type with growth indicator h(θ) and completely regular growth, which satisfies the following conditions:
1) All the zeros λ1,λ2,… of L(λ) are simple, and |λn+1|−|λn|⩾.
2) We have the estimate
\bigl|L(re^{i\theta})\bigr|<\frac{e^{h(\theta)r}}{\varphi(r)},\qquad r>r_0.
3) The sequence \{\lambda_n\} is part of a sequence \{\mu_n\}, \lim_{n\to\infty}\frac n{|\mu_n|}<\infty, which depends on the region D but not on the function \varphi(r). In this paper we prove an analogous theorem for entire functions of arbitrary finite order \rho.
Bibliography: 6 titles.
\Bibitem{Leo69}
\by A.~F.~Leont'ev
\paper On~the representation of analytic functions by Dirichlet series
\jour Math. USSR-Sb.
\yr 1969
\vol 9
\issue 1
\pages 111--150
\mathnet{http://mi.mathnet.ru/eng/sm3608}
\crossref{https://doi.org/10.1070/SM1969v009n01ABEH002048}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=277697}
\zmath{https://zbmath.org/?q=an:0183.34103|0198.10802}
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This publication is cited in the following 10 articles:
A. P. Khromov, “Finite-dimensional perturbations of Volterra operators”, Journal of Mathematical Sciences, 138:5 (2006), 5893–6066
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
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
Yu. I. Mel'nik, “The expansion of analytic functions in Dirichlet series”, Ukr Math J, 27:6 (1976), 672
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
V. K. Dzyadyk, “On convergence conditions for Dirichlet series on closed polygons”, Math. USSR-Sb., 24:4 (1974), 463–481
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
A. F. Leont'ev, “On conditions of expandibility of analytic functions in Dirichlet series”, Math. USSR-Izv., 6:6 (1972), 1265–1277
A. F. Leont'ev, “On methods of solution of an infinite order equation in the real
domain”, Math. USSR-Izv., 4:4 (1970), 859–890
A. F. Leont'ev, “On the representation of analytic functions in an open region by Dirichlet series”, Math. USSR-Sb., 10:4 (1970), 503–530