Multiple interpolation and principal parts of a Laurent series for meromorphic functions in the unit disk with power growth of the Nevanlinna characteristic
Let us fix the following notations: \(D\subset \mathbb{C}\) is the open unit disk, \(H(D)\) is the space of holomorphic functions in \(D\), \(M(D)\) is the set of all meromorphic functions in \(D\). For \(a>0\), denote \[ S^{\infty}_{a}=\left \{f\in M(D) ; \;T(r, f)\leq\frac{C_{f}}{(1-r)^{a}}\right \}, \] where for any \(r\in [0, 1)\), \(T(r, f)\) denotes the Nevanlinna characteristic of \(f\), \(S_{a, a}^{\infty}=S_{a}^{\infty}\cap H(D)\), the angle with vertex \(e^{i \theta}\) contained in \(D\), having opening \(\pi \delta\), with \(0 <\delta <1\), and the bisector \(r e^{i\theta}\) is said to be the Stolz angle and denoted by \(\Gamma_{\delta}(\theta)\). \(\tilde{\Delta}\) is the space of the sequences \((z_{j})_{j\in \mathbb{N}}\subset D\), with the properties \[ n(r)=\mathrm{card}\{z_{k} ; |z_{k}|\leq \frac{c}{(1-r)^{a+1}}\},\quad |\pi^{\prime}_{p, n}(z_{n}, z_{j})|\geq \exp(-c_{0}/(|1-|z_{n}|)^{a+1}), \] \[ c_{0}>0\quad\text{and}\quad\sup_{k\geq 1}\{p_{k}\}=n, \] where \(\pi_{p, n}(z, z_{j})\) is the so-called Djrbashian infinite product with zeros at the points of the sequence \((z_{k})_{k\in \mathbb{N}}\) without \(n\)-th factor (given by Formula (2.1) in the paper). The main result of the paper is the following. {Theorem.} Let \((\alpha_{k})_{k}\subset \bigcup_{s=1}^{n}\Gamma_{\delta}(\theta_{s})\) for certain \(\delta\) with \(0<\delta <\frac{1}{a+1}\). If \((\alpha_{k})_{k\in \mathbb{N}}\in \tilde{\Delta}\), then, for any sequence \((\gamma_{k})_{k}\) satisfying the condition \[ |\gamma_{k}|\leq \exp(-c_{0}/(|1-|z_{n}|)^{a+1}),\quad k\in \mathbb{N}, \quad c_{0}>0, \] it is possible to construct a function \(f\in S_{a, a}^{\infty}\) which solves the multiple interpolation problem \(f^{(s_{k}-1)}(\alpha_{k})=\gamma_{k}\), \(k\in \mathbb{N}\), for all \(s_{k}\geq 1\). Conversely, if the above multiple interpolation problem is solvable for all \(s_{k}\geq 1\) and \((\gamma_{k})_{k}\), with the above upper estimates, then the interpolation nodes \((\alpha_{k})_{k\in \mathbb{N}}\) belong to \(\tilde{\Delta}\).
scientific article; zbMATH DE number 6692076