New findings on the Bank-Sauer approach in oscillation theory
type: Item
instance of: scholarly article
title: New findings on the Bank-Sauer approach in oscillation theory
zbMATH Open document ID: 1256.34080
author: Janne Gröhn
author: Janne Heittokangas
published in: Constructive Approximation
publication date: 2012-06-26T00:00:00Z
review text: Consider the linear differential equation \[ f'' + a(z)f=0 ,\tag{1} \] where \(a\) is an analytic function in the unit disc \(D\) or in the complex plane. It was proved by \textit{V. Šeda} [Acta Fac. Rer. natur. Univ. Comenian., Math. 4, 223--253 (1959; Zbl 0099.06603)] that, given a sequence of distinct complex numbers \((z_n)\) with no finite limit points, there exists an entire function \(a\) such that the equation (1) has an entire solution \(f\) with the zero sequence~\((z_n)\). This result was recently generalized for an arbitrary domain \(G\) when the condition \(f(z_n)=0\) is replaced by \(f(z_n)=b_n\), \((b_n)\) being an arbitrary sequence [\textit{B. Vynnyts'kyi} and \textit{O. Shavala}, Acta Math. Univ. Comen., New Ser. 81, No. 1, 55--60 (2012; Zbl 1265.34325)]. The authors search for a description of zero sequences \((z_n)\) of solutions of (1) when \(a\) belongs to some growth class, namely, \(a\) is of finite order or from the Korenblum space, etc. Pioneering results in this direction for the case of the complex plane belong to \textit{S. B. Bank} [Result. Math. 13, No. 1--2, 1--11 (1988; Zbl 0662.34007)] who found a general necessary condition for \((z_n)\) in the case when \(a\) is of finite order, i.e., \(\log \log M(r,a)=O(\log r)\) \((r\to+\infty)\), where \(M(r,a)=\max \{|f(z)|: |z|=r\}\). On the other hand, general sufficient conditions for the complex plane were obtained by \textit{A. Sauer} [Proc. Am. Math. Soc. 125, No. 4, 1143--1147 (1997; Zbl 0866.34002)], and then improved by \textit{J. Heittokangas} and \textit{I. Laine} [Acta Math. Univ. Comen., New Ser. 74, No. 2, 287--307 (2005; Zbl 1164.34562)]. Let \({\mathcal A}^{-q}\), \(q\geq 0\), denote the space of analytic functions \(g\) in the unit disc such that \( M(r,g)=O((1-r)^{-q})\), \(r\in (0,1)\). Let \(\sigma(z, w)=|\frac{z-w}{1-\bar z w}|\). A sequence \((z_n)\) is called uniformly \(q\)-separated in \(D\) if there is a constant \(q\geq 0\) such that \(\inf_{k} (1-|z_k|)^{- q} \prod…
Mathematics Subject Classification ID: 34M10
Mathematics Subject Classification ID: 30J10
Mathematics Subject Classification ID: 34M03
zbMATH DE Number: 6050528
zbMATH Keywords: Blaschke-oscillatory
zbMATH Keywords: Blaschke product
zbMATH Keywords: logarithmic derivative
zbMATH Keywords: oscillation theory
zbMATH Keywords: prescribed zero sequence
zbMATH Keywords: uniformly separated sequence
zbMATH Keywords: Korenblum's space
zbMATH Keywords: interploation
reviewed by: Igor Chyzhykov
MaRDI profile type: Publication
full work available at URL: s00365-011-9137-8
OpenAlex ID: W1996239400
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DOI: 10.1007/S00365-011-9137-8
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description: scientific article; zbMATH DE number 6050528