Geodesic
Convergence of series of exponential monomials
Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 56-68

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In the paper we study the convergence of series of exponential monomials, special cases of which are the series of exponentials, Dirichlet series and power series. We provide a description of the space of coefficients of series of exponential monomials converging in a given convex domain in the complex plane is described. Under a single natural restriction on the degrees of monomials, we provide a complete analogue of the Abel theorem for such series, which, in particular, implies results on the continued convergence of series of exponential monomials. We also obtain a complete analogue of the Cauchy-Hadamard theorem, in which we give a formula allowing to recover the convergence domain of these series by their coefficients. The obtained results include, as special cases, all previously known results related with the Abel and Cauchy-Hadamard theorems for exponential series, Dirichlet series and power series.
Keywords: exponential monomial, Abel theorem, Cauchy-Hadamard theorem.
Mots-clés : convex domain 
A. S. Krivosheev; O. A. Krivosheeva. Convergence of series of exponential monomials. Ufa mathematical journal, Tome 14 (2022) no. 4, pp. 56-68. http://geodesic.mathdoc.fr/item/UFA_2022_14_4_a4/
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[1] A.F. Leontiev, Entire functions. Exponential series, Nauka, M., 1983 (in Russian) | MR

[2] A.F. Leontiev, Exponential series, Nauka, M., 1976 (in Russian) | MR | MR

[3] E. Hille, “Note on Dirichlet's series with complex exponents”, Ann. of Math., 25 (1924), 261–278 | DOI | MR

[4] G.L. Lunts, “On some generalizations of Dirichlet series”, Matem. Sborn., 10(52):1–2 (1942), 35–50 (in Russian)

[5] G.L. Lunts, “On one class of generalized Dirichlet series”, Uspekhi Matem. Nauk., 12:3(75) (1957), 173–179 (in Russian) | MR | Zbl

[6] A.V. Bratischev, Köther bases, entire functions and their applications, Habilitation thesis, Rostov-on-Don, 1995 (in Russian)

[7] O.A. Krivosheyeva, “The convergence domain for series of exponential monomials”, Ufa Math. J., 3:2 (2011), 42–55 | MR | Zbl

[8] K. Liechtweiss, Konvexe Mengen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1980 (in German) | MR | MR