On the divergence of trigonometric Fourier series in classes \(\varphi(L)\) close to \(L\) (Q1675317)

Let \(\varphi:[0,\infty)\to[0,\infty)\) denote a nondecreasing function such that \(\lim_{u\to\infty}\varphi(u)=\infty\) and let \(\varphi(L)\) denote the set of all \(2\pi\)-periodic functions measurable on \([0,2\pi)\) such that \(\int\varphi(| f(t)| )\,dt<\infty\). In particular, \(L^p\) is the set \(\varphi(L)\) for \(\varphi(t)=t^p\). Let \(S_n(f,x)\), \(n\geq 1\), denote the partial Fourier sums for a \(2\pi\)-periodic function \(f\) integrable on \([0,2\pi)\). Riesz showed that if \(p>1\) and \(f\in L^p\), then \(\int_0^{2\pi}| S_n(f,t)-f(t)| ^pdt\to 0\) as \(n\to\infty\). Kolmogorov proved that the same convergence hold for \(0<p<1\). In the paper under the review the author generalizes these results in the following way: For a function \(\chi:[0,\infty)\to[0,\infty)\) denote \(\varphi(u)=u\int_1^u\chi(t)t^{-2}\,dt\) for \(u>1\) and \(\varphi(u)=0\) for \(0\leq u\leq 1\). Assume that the function \(\chi:[0,\infty)\to[0,\infty)\) has these properties: \(\chi(u)/u^s\) is nondecreasing for some \(s>0\), \(\chi(0)=0\), \(\chi(2u)\) is \(O(\chi(u))\) and \(\chi(u)\to\infty\) as \(u\to\infty\). Then \(\int_0^{2\pi}\chi(| S_n(f,t)-f(t)| )\,dt\to0\) as \(n\to\infty\) for every \(f\in\varphi(L)\). The author shows that this theorem does not hold, if the function \(\chi\) grows slower that any \(t^p\) with \(p>1\).