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Krivosheev A. S., Krivosheeva O. A. Representation of functions on a line by a series of exponential monomials. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2022, vol. 22, iss. 4, pp. 416-429. DOI: 10.18500/1816-9791-2022-22-4-416-429, EDN: TPUWZW

# Representation of functions on a line by a series of exponential monomials

In this work, we consider the weight spaces of integrable functions $L_p^\omega$ ($p\geq 1$) and continuous functions $C^\omega$ on the real line. Let $\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers $\lambda_k$ and their multiplicities $n_k$, $\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence $\Lambda$. We study the subspaces $W^p (\Lambda,\omega)$ and $W^0 (\Lambda,\omega)$, which are the closures of the linear span of the system $\mathcal{E}(\Lambda)$ in the spaces $L_p^\omega$ and $C^\omega$, respectively. Under natural constraints on $\Lambda$ (the finiteness of the condensation index $S_\Lambda$ and $n_k/\lambda_k\leq c$, $k\geq 1$) and on the convex weight $\omega$, conditions are obtained under which each function of these subspaces continues to an entire function and is represented by a series in the system $\mathcal{E}(\Lambda)$ that converges absolutely and uniformly on compact sets in the plane. In contrast to the previously known results for the specified representation problem, we do not require that the sequence $\Lambda$ has a density, and we do not impose the separability condition: $\lambda_{k+1}-\lambda_k\geq h$, $k\geq 1$ (instead, the condition of equality to zero of the special condensation index is used).

- Anderson J. M., Binmore K. G. Closure theorems with applications to entire functions with gaps.
*Transactions of the American Mathematical Society*, 1971, vol. 161, pp. 381–400. https://doi.org/10.2307/1995948 - Deng G. T. Incompleteness and closure of a linear span of exponential system in a weighted Banach space.
*Journal of Approximation Theory*, 2003, vol. 125, iss. 1, pp. 1–9. https://doi.org/10.1016/j.jat.2003.09.004 - Zikkos E. The Closed span of some exponential system in weighted Banach spaces on the real line and a moment problem.
*Analysis Mathematica*, 2018, vol. 44, iss. 4, pp. 605–630. https://doi.org/10.1007/s10476-018-0311-0 - Krivosheev A. S., Krivosheeva O. A., Kuzhaev A. F. The representation by series of exponential monomials of functions from weight subspaces on a line.
*Lobachevskii Journal of Mathematics*, 2021, vol. 42, iss. 6, pp. 1183–1200. https://doi.org/10.1134/S1995080221060159 - Boas R. P. Jr.
*Entire Functions*. New York, Academic Press, 1954. 276 p. - Krivosheev A. S. A fundamental principle for invariant subspaces in convex domains.
*Izvestiya: Mathematics*, 2004, vol. 68, iss. 2, pp. 291–353. https://doi.org/10.1070/IM2004v068n02ABEH000476 - Krivosheev A. S., Krivosheeva O. A., Rafikov A. I. Invariant subspaces in half-plane.
*Ufa Mathematical Journal*, 2021, vol. 13, iss. 3, pp. 57–79. https://doi.org/10.13108/2021-13-3-57 - Levin B. Ja.
*Distribution of Zeros of Entire Functions.*Providence, American Mathematical Society, 1964. 583 p. (Russ. ed.: Moscow, Gostekhizdat, 1956. 632 p.). - Leont’ev A. F.
*Tselye funktsii. Ryady eksponent*[Entire Functions. Series of Exponentials]. Moscow, Nauka, 1983, 176 p. (in Russian). - Rockafellar R. T.
*Convex Analysis*. New Jersey, Princeton University Press, 1970. 470 p.

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