On locally bounded solutions of Schilling's problem
type: Item
instance of: scholarly article
title: On locally bounded solutions of Schilling's problem
zbMATH Open document ID: 0980.39013
author: Janusz Morawiec
published in: Annales Polonici Mathematici
publication date: 2001-07-09T00:00:00Z
Mathematics Subject Classification ID: 39B12
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 1615687
zbMATH Keywords: Schilling's problem
zbMATH Keywords: functional equations
zbMATH Keywords: locally bounded solutions
MaRDI profile type: Publication
full work available at URL: ap76-3-1
OpenAlex ID: W1978945347
DOI: 10.4064/AP76-3-1
review text: A solution to Schilling's problem is a function \(f:\mathbb{R} \to \mathbb{R} \) which, for given \(q\in(0,1)\), satisfies the functional equation NEWLINE\[NEWLINE4qf(x)=f(x-1)+2f(x)+f(x+1), \qquad x\in \mathbb{R} , NEWLINE\]NEWLINE and which vanishes outside of the interval \([-q/(1-q),q/(1-q)]=:[-Q,Q]\). The author proves the following two results: (1) If \(q=(3-\sqrt{5})/2\), then every non-trivial solution is unbounded around any point \(x\in[-Q,Q]\). (2) The same conclusion holds for any \(q\in (1/3,1/2)\) which satisfies an equation of the form \(2\sum_{k=1}^K q^k + \lambda q^{K+1}=1\) where \(K\) is a positive integer and \(\lambda \in \{1,2\}\).
reviewed by: Roland Girgensohn
description: scientific article; zbMATH DE number 1615687