Divergence almost everywhere of Fourier series with respect to the Ciesielski system. (Russian. English summary) Zbl 0686.42021

Summary: Estimates from below for the Franklin functions are obtained. With the help of these estimates, it is proved, that for some uniformly bounded complete orthonormal spline systems (included the Ciesielski system), there exist Fourier series diverging almost everywhere. As a consequence of a theorem proved earlier by one of the authors, it follows, that after removing a finite number of functions from these systems it is impossible multiplicatively to complete the remaining system up to the basis in \(L^ p_{[0,1]}\), \(1\leq p<\infty\).


42C15 General harmonic expansions, frames
42A20 Convergence and absolute convergence of Fourier and trigonometric series