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Bounded solutions of Schilling’s problem. (English) Zbl 0860.39031

The author considers Schilling’s functional equation \[ f(qx) = {1\over 4q} \bigl(f(x-1) + f(x+1) + 2f(x) \bigr), \] for functions \(f:R \to R\) such that \(f(x)=0\) for \(|x |> q/(1-q)\), where \(q\) is a fixed number in (0,1). Fixing an integer \(n\) and \(q\) in \((0,q_n]\) where \(q_n\) is the unique solution in \((1/3,1/2)\) of \(x^{n+1} - 3x+1 = 0\), it is proved that there is a set \(A_q^n\) in the real line such that any solution \(f\) of Schilling’s functional equation which is bounded in a neighbourhood of a point of \(A^n_q\) vanishes everywhere. Sets \(A^n_q\) and their union are studied in detail.

MSC:

39B22 Functional equations for real functions
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