Matrix transformations from absolutely convergent series to convergent sequences as general weighted mean summability methods (Q5926157)

A series \(\sum^\infty_{k=0}x_k\) of complex numbers is said to be summable \((C,1)\) if the sequence NEWLINE\[NEWLINE{1\over n+1}\sum^n_{k=0} \sum^k_{i=0} x_i,\quad n=0,1,2,\dots\tag{1}NEWLINE\]NEWLINE converges to a finite limit as \(n\to\infty\) and is said to be summable \(\overline N\) if the sequence NEWLINE\[NEWLINE{1\over P_n} \sum^n_{k=0} p_k\sum^k_{i =0} x_i,\quad n=0,1,2,\dots NEWLINE\]NEWLINE converges to a finite limit as \(n\to\infty\) where \(\overline N\) is a weighted mean triangular matrix with entries NEWLINE\[NEWLINEa_{n,k}= {p_k \over P_n}, \quad k=0,1,2, \dots,n, \quad n=0,1,2, \dots,\quad P_n=\sum^n_{k=0} p_k.NEWLINE\]NEWLINE \textit{G. H. Hardy} [A theorem concerning summable series, Cambr. Phil. Soc. Proc. 20, 304-307 (1921; JFM 48.1190.01)] proved a necessary and sufficient condition for the series (1) to be \((C,1)\) summable to a finite number \(L\). Móricz and Rhoades established necessary and sufficient conditions for (1) to be \(\overline N\) summable to a finite number \(L\). In the present paper the author gives necessary and sufficient conditions for an infinite matrix \(A=(a_{n,k})\) which is a general weighted mean to be a mapping from absolutely convergent series to convergent sequences. Hardy's and Móricz and Rhoades's results follow as corollary from the present result.