Divergent Fourier series of integrable functions with respect to orthonormal systems (Q1280633)

Given two orthonormal systems, \(\varphi^1\) and \(\varphi^2\), let \(E^1(f)\) and \(E^2(f)\) denote the sets of point of divergence of the Fourier series of a function \(f \in L[0,1]\) with respect to \(\varphi^1\) and \(\varphi^2\), respectively. The author establishes existence of orthonormal systems \(\varphi^1\) and \(\varphi^2\) such that for every \(f \in L[0,1]\), \(f \neq 0\), \[ E^1(f) \cup E^2(f)=[0,1] . \] In particular, at least one of the corresponding Fourier series diverges on a set of positive measure.