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JOURNALS // Eurasian Mathematical Journal // Archive

Eurasian Math. J., 2022 Volume 13, Number 2, Pages 37–42 (Mi emj436)

This article is cited in 2 papers

Completeness of the exponential system on a segment of the real axis

A. M. Gaisina, B. E. Kanguzhinbc, A. A. Seitovabc

a Institute of Mathematics with Computing Centre, Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, Bashkir State University, 112 Chernyshevsky St, 450008 Ufa, Russia
b Institute of Mathematics and Mathematical Modeling, 125 Pushkin St, 050010 Almaty, Kazakhstan
c Al-Farabi Kazakh National University, 71 al-Farabi Ave, 050040 Almaty, Kazakhstan

Abstract: Let $\Lambda=\{\lambda_n\}$ be the sequence of all zeros of the entire function $\Delta(\lambda)=1-i\lambda\int_0^1f(t)e^{i\lambda t}dt$ of exponential type. We consider exponential system of functions $e(\Lambda)=\{t^{p-1}e^{i\lambda_nt}, 1\leqslant p\leqslant m_n\}$, where $m_n$ — is the multiplicity of the zero $\lambda_n$. The question is: for which $a$$b$ ($a<b$) is the system $e(\Lambda)$ complete (incomplete) in the space $L^2(a, b)$? Let $D$ be the length of the indicator conjugate diagram of the entire function $\Delta(\lambda)$. Then the following statements are valid:

Keywords and phrases: Lebesgue-Stieltjes integral, indicatrix of the growth, Borel adjoint diagram, Beurling-Malliavin multiplier theorem, Paley-Wiener theorem, Cartwright class.

MSC: 30D15, 30D20, 46E30

Received: 21.02.2021

Language: English

DOI: 10.32523/2077-9879-2022-13-2-37-42



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