On the divergence of strong \(\Phi\)-means of Fourier series
The author completes the proof of Totik's hypothesis on strong summability of Fourier series. He proves the following theorem: Let \(\Phi\) be a continuous increasing function with \(\Phi(0)= 0\) for which \(\lim_{t\to \infty} (\log \Phi(t))/t= \infty\). Then there exists \(f\in L^1(0, 2\pi)\) such that \[ \lim_{n\to \infty} {1\over n} \sum^n_{k= 1} \Phi(|S_k(x, f)|)= \infty\quad \text{a.e.}, \] where \(S_k(x, f)\) is the partial sum of the Fourier series of \(f\).
scientific article; zbMATH DE number 751097