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 Ufimsk. Mat. Zh., 2019, Volume 11, Issue 1, Pages 19–25 (Mi ufa457)

On an interpolation problem in the class of functions of exponential type in a half-plane

B. V. Vynnyt'skyi, V. L. Sharan, I. B. Sheparovych

Drohobych state pedagogical university named after Ivan Franko, Stryiskaya str., 3, 82100, Drohobych, Ukraine

Abstract: Solvability conditions for interpolation problem $f(n)=d_{n},\quad n \in {\mathbb{N}}$ in the class of entire functions satisfying the condition $| {f(z)} |\le e^{\pi | {\mathrm{Im}ż |+o( {| z |} )}, z\to \infty$ are well known. In the presented paper we study the interpolation problem $f(\lambda_ {n}) = d_ {n}$ in the class of exponential type functions in the half-plane. We find sufficient solvability conditions for the considerate problem. In particular, a sufficient part of Carleson's interpolation theorem is generalized and an analogue of a classic interpolation condition is found in the form
$$\sum\limits_{j = k}^{\infty} \mathrm{Re} ( - \xi _{j} \frac{\lambda _{k} ^{2} - 1}{\lambda _{k} + \overline {\lambda_j}} ) \le c_{3}, \qquad \xi _{j} : = \frac{\mathrm{Re} \lambda_j} {1 + | \lambda_j|^{2}}.$$
The necessity of sufficient conditions is also discussed. The results are applied to studying a problem on splitting and searching an analogue of the identity $2\cos z=\exp(-iz)+\exp(iz)$ for each function of exponential type in the half-plane. We prove that each holomorphic in the right-hand half-plane function $f$ obeying the , estimate $| {f(z)} |\le O(\exp(\sigma| \mathrm{Im}ż|))$ can be represented in the form $f=f_1+f_2$ and the functions $f_1$ and $f_2$ holomorphic in the right-hand half-plane satisfy conditions
$$| {f_1(z)} |\le O (\exp(| z|h_{-}(\varphi)))\quadand | {f_2(z)} |\le O(\exp(| z|h_{+}(\varphi))),$$
where $\sigma\in [0;+\infty)$, $z = re^{i\varphi}$,
h_{ +} (\varphi ) = \{ \begin{aligned} &\sigma {| {\sin \varphi} |}, && \varphi \in [0;\frac{\pi}{2}], &0, &&\varphi \in [-\frac{\pi}{2};0], \end{aligned}. \qquad h_{ -} (\varphi ) = \{ \begin{aligned} &0, &&\varphi \in [0;\frac{\pi}{2}], &\sigma {| {\sin \varphi} |}, && \varphi \in [ -\frac{\pi}{2};0]. \end{aligned}.
The paper uses methods works by L. Carleson, P. Jones, K. Kazaryan, K. Malyutin and other mathematicians.

Keywords: holomorphic functions of exponential type in the half-plane, interpolation, splitting of holomorphic functions.

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English version:
Ufa Mathematical Journal, 2019, 11:1, 19–26 (PDF, 338 kB); https://doi.org/10.13108/2019-11-1-19

Bibliographic databases:

UDC: 517.5
MSC: 30E05, 30D15

Citation: B. V. Vynnyt'skyi, V. L. Sharan, I. B. Sheparovych, “On an interpolation problem in the class of functions of exponential type in a half-plane”, Ufimsk. Mat. Zh., 11:1 (2019), 19–25; Ufa Math. J., 11:1 (2019), 19–26

Citation in format AMSBIB
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