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

Abstract

Solvability conditions for interpolation problem $f(n)=d_{n},\quad n \in {\mathbb{N}} $ in the class of entire functions satisfying the condition $ \left| {f(z)} \right|\le e^{\pi \left| {\mathrm{Im}\,z} \right|+o\left( {\left| z \right|} \right)}, 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 $$\sum\limits_{j = k}^{\infty} \mathrm{Re}\,\left( - \xi _{j} \frac{\lambda _{k} ^{2} - 1}{\lambda _{k} + \overline {\lambda_j}} \right) \le c_{3}, \qquad \xi _{j} : = \frac{\mathrm{Re}\,\lambda_j} {1 + \left| \lambda_j\right|^{2}}.$$ 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 $\left| {f(z)} \right|\le O(\exp(\sigma| \mathrm{Im}\,z|))$ 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 $$ \left| {f_1(z)} \right|\le O (\exp(| z|h_{-}(\varphi)))\quad\text{and} \left| {f_2(z)} \right|\le O(\exp(| z|h_{+}(\varphi))), $$ where $\sigma\in [0;+\infty)$, $z = re^{i\varphi}$, $$h_{ +} (\varphi ) = \left\{ \begin{aligned} &\sigma {\left| {\sin \varphi} \right|}, && \varphi \in \left[0;\frac{\pi}{2}\right], \\ &0, &&\varphi \in \left[-\frac{\pi}{2};0\right], \end{aligned}\right. \qquad h_{ -} (\varphi ) = \left\{ \begin{aligned} &0, &&\varphi \in \left[0;\frac{\pi}{2}\right], \\ &\sigma {\left| {\sin \varphi} \right|}, && \varphi \in \left[ -\frac{\pi}{2};0\right]. \end{aligned}\right. $$ The paper uses methods works by L. Carleson, P. Jones, K. Kazaryan, K. Malyutin and other mathematicians.