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Stability of zero (sub)sequences for classes of holomorphic functions of moderate growth in the unit disk. (Russian. English summary) Zbl 1249.30012

Summary: Let \(\Lambda=\{\lambda_{k}\}\) and \(\Gamma=\{\gamma_{k}\}\) be two sequences of points in the unit disk \(\mathbb D=\{z\in \mathbb C :| z|<1\}\) of the complex plane \(\mathbb C\), and let \(H\) be a weight space of holomorphic functions on \(\mathbb D\). Suppose that \( \Lambda\) is the zero subsequence of some nonzero function from \(H\). The authors give conditions of closeness of the sequence \( \Gamma\) to the sequence \( \Lambda,\) under which the sequence \( \Gamma\) is the zero sequence for some holomorphic function from a space \(\widehat{H}\supset H\). The space \(\widehat{H}\) can be a little larger than \(H\).

MSC:

30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral)