On continuous solutions of a problem of R. Schilling (Q1895250)

It is proved that if \(q\) is one of the numbers \((\sqrt 3 - 1)/2\), \((3 - \sqrt 5)/2\), \(\sqrt 2 - 1\), \((\sqrt 5 - 1)/2\), and \(f : \mathbb{R} \to \mathbb{R}\) satisfies \(4qf(qx) = f(x - 1) + f(x + 1) + 2f (x)\) for \(x \in \mathbb{R}\) and \(f(x) = 0\) for \(|x |> q/(1-q)\), then \(f\) vanishes on \(\mathbb{Z} + q \mathbb{Z}\).