An extension of the Sidon-Fomin type inequality and its applications (Q2720284)

The author presents new convergence and integrability classes for cosine series, relying on the following inequality. There exists an absolute constant \(M\) such that if \(\{a_k: k=0,1,\dots\}\) is a sequence of real numbers for which \(|a_k|\leq 1\) for all \(k\) and if \(n\), \(r= 0,1,2,\dots\), then NEWLINE\[NEWLINE\int^\pi_{-\pi} \Biggl|\sum^n_{k= 0} a_k D^{(r)}_k(x)\Biggr|dx\leq M(n+ 1)^{r+ 1},NEWLINE\]NEWLINE where \(D^{(r)}_n\) is the \(r\)th derivative of the Dirichlet kernel. The case \(r=0\) was proved by \textit{S. Sidon} [J. London Math. Soc. 14, 158-160 (1939; Zbl 0021.40301)] and reproved by \textit{G. A. Fomin} [Mat. Sb., n. Ser. 5(107), 144-152 (1964; Zbl 0133.02403)]. The case \(r\geq 1\) follows from the case \(r=0\) via Bernstein's inequality.