On the newly generalized absolute Cesàro summability of orthogonal series (Q1897511)

The author has proved three general theorems which generalize many well-known own results as well as results of G. Sunouchi, I. Szalay, and K. Tandori. One of them is the following: Theorem 1. Let us assume that \(\alpha> 1/2\), \(1\leq k\leq 2\) and \(\gamma(t)\) is a positive nondecreasing function on \([1, \infty)\) such that the sequence \(\gamma(n)\) is quasi \(\eta\)-power-monotone decreasing with some \(\eta> - 1\). Then the condition \[ \sum^\infty_{m= 0} \gamma(2^m)^k \Biggl\{ \sum^{2^{m+ 1}}_{n= 2^m+ 1} c^2_n\Biggr\}^{k/2}< \infty \] is necessary and sufficient for the series \(\sum^\infty_{n= 0} c_n \varphi_n(x)\) to be summable \(|C, \alpha, \gamma(t)|_k\) for every orthogonal system \(\{\varphi_n(x)\}\) almost everywhere in \((0, 1)\).