On locally bounded solutions of Schilling's problem (Q2724129)

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 \(\lambda \in \{1,2\}\).