Article
Ufa Mathematical Journal
Volume 11, Number 1, pp. 16-26
On an interpolation problem in the class of functions of exponential type in a half-plane
Sharan V.L., Sheparovych I.B., Vynnyts'kyi B.V.
DOI:10.13108/2019-11-1-16
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olvability 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.