Interpolation by Jones-Type Series in Spaces of Analytic Functions in the Half-Plane

6 Jul 2021, 10:00


Konstantin Malyutin (Kursk State University, Russia)


Denote by $\mathcal{A}_+$ the space of analytic functions in the upper half-plane $\mathbb{C}_+=\{z:\Im z>0\}$, where $\mathcal{A}_+$ is one of the spaces: 1) the space functions of finite order $\rho>1$, 2) the space functions of finite order $\rho>0$ and of normal type, 3) the space of bounded functions $H^\infty$.
Let $D=\{a_n,q_n\}_{n=1}^\infty,$ be a divisor (i.e., a set of distinct complex numbers $\{a_n\}_{n=1}^\infty\subset\mathbb{C}_+$ with limit points on the real axis, together with their integer multiplicities $\{q_n\}_{n=1}^\infty\subset\mathbb N$). In the space
$\mathcal{A}_+$, the interpolation problem is considered:
$$ F^{(k-1)}(a_n)=b_{n,k},\quad k=1,2,\dots,q_n,\> n\in\mathbb N\,\quad F\in\mathcal{A}_+. $$ We find a criterion for the interpolation of the divisor $D$ in the spaces $\mathcal{A}+$ in terms of canonical products and in terms of the Nevanlinna measure $\mu+(G)=\sum_{a_n\in G}q_n\sin(\arg a_n)$ defined by the interpolation nodes.
The solution to the problem is constructed in the form of a Jones-type interpolation series.
This method is also used to solve interpolation problems in spaces of meromorphic functions in a half-plane with a given growth.

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Konstantin Malyutin (Kursk State University, Russia)

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