Higher-order linear differential equations with solutions having a prescribed sequence of zeros and lying in the Dirichlet space

Li-peng Xiao¹

¹ Institute of Mathematics and Information Jiangxi Normal University Nanchang, 330022, China

Annales Polonici Mathematici, Tome 115 (2015) no. 3, pp. 275-295

Résumé

The aim of this paper is to consider the following three problems:(1) for a given uniformly $q$-separated sequence satisfying certain conditions, find a coefficient function $A(z)$ analytic in the unit disc such that $f'''+A(z)f=0$ possesses a solution having zeros precisely at the points of this sequence;(2) find necessary and sufficient conditions for the differential equation $$ f^{(k)}+A_{k-1}f^{(k-1)}+\cdots+A_1f'+A_0f=0\tag*{$(*)$} $$ in the unit disc to be Blaschke-oscillatory;(3) find sufficient conditions on the analytic coefficients of the differential equation $(*)$ for all analytic solutions to belong to the Dirichlet space $\mathcal{D}$. Our results are a generalization of some earlier results due to J. Heittokangas and J. Gröhn.

Détail

DOI : 10.4064/ap115-3-6

Keywords: paper consider following three problems given uniformly q separated sequence satisfying certain conditions coefficient function analytic unit disc f possesses solution having zeros precisely points sequence necessary sufficient conditions differential equation k k cdots tag* * unit disc blaschke oscillatory sufficient conditions analytic coefficients differential equation nbsp * analytic solutions belong dirichlet space mathcal results generalization earlier results due nbsp heittokangas nbsp