On unbounded divergence sets of series in orthonormal bases (Q1931623)

Let \(\{\psi_n(x)\}_{n=1}^\infty\), \(0\leq x \leq 1\), be an orthogonal system of continuous functions. Assuming that \(\{\psi_n\}_{n=1}^\infty\) has a localization property, the author proves the following theorem: For every \(G_\delta\)-set \(E\subset [0,1]\), there exists a series \[ \sum_{n=1}^\infty a_n \psi_n(x) \] that unboundedly diverges for \(x\in E\) and converges for \(x\notin E\).