On the \((C,-1<\alpha<0)\)-summability of series with respect to block-orthonormal systems (Q2702988)

Let \(\{N_k: k= 1,2,\dots\}\) be an increasing sequence of natural numbers and \(\Delta_k:= (N_k,N_{k+ 1}]\). A system \(\{\varphi_n:n= 1,2,\dots\}\) of functions in \(L^2(0,1)\) is called \(\Delta_k\)-orthonormal if \(\|\varphi_n\|_2= 1\) for all \(n\geq 1\), and \((\varphi_i, \varphi_j)= 0\) for all \(i,j\in \Delta_k\), \(i\neq j\), \(k\geq 1\).NEWLINENEWLINENEWLINEThe author proves three theorems of the Cesàro summability of series \(\sum a_n\varphi_n\), where \(\{\varphi_n\}\) is block-orthonormal.NEWLINENEWLINENEWLINEReviewer's remark: The notion of blockwise orthogonality was introduced and first studied by the reviewer [Proc. Am. Math. Soc. 101, 709-715 (1987; Zbl 0632.60025)].