New findings on the Bank-Sauer approach in oscillation theory
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 \(\inf_{k} (1-|z_k|)^{- q} \prod_{n\neq k} \sigma(z_n, z_k)>0\). This concept is close but different from the upper uniform density introduced by \textit{K. Seip} [Invent. Math. 113, No. 1, 21--39 (1993; Zbl 0789.30025)]. If \(q=0\), we obtain the definition of a uniformly separated sequence due to L. Carleson. Theorem 6. Let \((z_n)\) be a uniformly \(q\)-separated sequence of non-zero points in \(D\). (a) Suppose that \((z_n)\) satisfies the condition \(\sum_n (1-|z_n|)^\alpha<\infty\) for some \(\alpha\in (0,1)\). Then there exists a function \(a\in {\mathcal A}^{-2(1+\alpha+2q)}\) satisfying \[ \limsup _{|z|\to 1-} (1-|z|^2)^2 |a(z)|\geq 1 \eqno(2) \] such that (1) possesses a solution whose zero sequence is \((z_n)\). (b) Suppose that \((z_n) \) is a finite union of uniformly separated sequences in \(D\). Then there exists a function \(a\in {\mathcal A}^{-2(1+2q)}\) satisfying (2) such that (1) possesses a solution whose zero sequence is \((z_n)\). Also, the following unit-disc analog of a result of Bank holds. Theorem 5. Let \((\varepsilon_n)\) be an infinite sequence of real numbers satisfying \(0<\varepsilon_n<\exp\{-2^n\}\), \(n\in\mathbb N\). Let \((z_n)\) be the sequence defined by \(z_{2n-1}=a_n\), \(z_{2n}=a_n+\varepsilon_n\), where \(a_n=1-2^{-n}\). Then \((z_n)\) cannot be zero sequence of a solution of (1), where \(a\in {\mathcal A}^{-\infty}=\bigcup_{p\geq 0} {\mathcal A}^{-p}\).
scientific article; zbMATH DE number 6050528
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