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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 3, Pages 50–62 (Mi ufa385)

Representation of functions in locally convex subspaces of $A^\infty (D)$ by series of exponentials

K. P. Isaevab, K. V. Trounova, R. S. Yulmukhametovab

a Bashkir State University, Zaki Validi str. 32, 450074, Ufa, Russia
b Institute of Mathematics, Ufa Scientific Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia

Abstract: Let $D$ be a bounded convex domain in the complex plane, $\mathcal M_0=(M_n)_{n=1}^\infty$ be a convex sequence of positive numbers satisfying the “non-quasi-analyticity” condition:
$$\sum_n\frac {M_n}{M_{n+1}}<\infty,$$
$\mathcal M_k=(M_{n+k})_{n=1}^\infty$, $k=0,1,2,3,\ldots$ be the sequences obtained from the initial ones by removing first $k$ terms. For each sequence $\mathcal M_0=(M_n)_{n=1}^\infty$ we consider the Banach space $H(\mathcal M_0,D)$ of functions analytic in a bounded convex domain $D$ with the norm:
$$\|f\| ^2=\sup_n \frac 1{M_n^2}\sup_{z\in D}|f^{(n)}(z)|^2.$$
In the work we study locally convex subspaces in the space of analytic functions in $D$ infinitely differentiable in $\overline D$ obtained as the inductive limit of the spaces $H(\mathcal M_k,D)$. We prove that for each convex domain there exists a system of exponentials $e^{\lambda_nz}$, $n\in \mathbb{N}$, such that each function in the inductive limit $f\in \lim ind H(\mathcal M_k,D):=\mathcal H(\mathcal M_0,D)$ is represented as the series over this system of exponentials and the series converges in the topology of $\mathcal H(\mathcal M_0,D)$. The main tool for constructing the systems of exponentials is entire functions with a prescribed asymptotic behavior. The characteristic functions $L$ with more sharp asymptotic estimates allow us to represent analytic functions by means of the series of the exponentials in the spaces with a finer topology. In the work we construct entire functions with gentle asymptotic estimates. In addition, we obtain lower bounds for the derivatives of these functions at zeroes.

Keywords: analytic functions, entire functions, subharmonic functions, series of exponentials.

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English version:
Ufa Mathematical Journal, 2017, 9:3, 48–60 (PDF, 369 kB); https://doi.org/10.13108/2017-9-3-48

Bibliographic databases:

UDC: 517.5
MSC: 30B50, 30D20, 30D60

Citation: K. P. Isaev, K. V. Trounov, R. S. Yulmukhametov, “Representation of functions in locally convex subspaces of $A^\infty (D)$ by series of exponentials”, Ufimsk. Mat. Zh., 9:3 (2017), 50–62; Ufa Math. J., 9:3 (2017), 48–60

Citation in format AMSBIB
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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. K. P. Isaev, “On entire functions with given asymptotic behavior”, Probl. anal. Issues Anal., 7(25), spetsvypusk (2018), 12–30
2. K. P. Isaev, K. V. Trounov, R. S. Yulmukhametov, “Representing systems of exponentials in projective limits of weighted subspaces of $H(D)$”, Izv. Math., 83:2 (2019), 232–250
3. K. P. Isaev, “Predstavlyayuschie sistemy eksponent v prostranstvakh analiticheskikh funktsii”, Kompleksnyi analiz. Tselye funktsii i ikh primeneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 161, VINITI RAN, M., 2019, 3–64
4. K. P. Isaev, K. V. Trunov, R. S. Yulmukhametov, “Predstavlenie ryadami eksponent funktsii v normirovannykh podprostranstvakh $A^\infty (D)$”, Kompleksnyi analiz. Matematicheskaya fizika, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 162, VINITI RAN, M., 2019, 42–56
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