On the series methods containing the method \(V_\sigma\) (Q2708393)

Let \(\sigma:\mathbb{N} \to\mathbb{N}\) be a one-to-one mapping such that \(\sigma^q (n)\neq n\) for all \(q,n=1,2, \dots\) and let \(V_\sigma: =\{x\in l^\infty \mid\lim_q t_{qn}(x)= \sigma\)-\(\lim x\) uniformly in \(n\}\), where \(t_{qn}(x): =(x_n +(Tx)_n+ \cdots+(T^qx)_n)/ (q+1)\) for \(q=0,1,\dots, n=1,2,\dots\), and \(Tx:= (x_{\sigma (n)})\). The main result of the paper is the following theorem: A regular series-to-sequence matrix method \(B=(b_{nk})\) contains the method \(V_\sigma\) (i.e. \(B\) sums every series \(\sum_kx_k\) such that \(s:=(\sum^n_{k=1} x_k)_n \in V_\sigma\) to the value \(\sigma\)-\(\lim s)\) if and only if (i) \(\lim_k b_{nk}= 0\) \((n=1,2, \dots)\) and (ii) \(\lim_n \sum_k|\Delta(b_{nk}-b_{n, \sigma (k)})|=0\) where \(\Delta (b_{nk}- b_{n,\sigma (k)}):= b_{nk}-b_{n,\sigma (k)} -b_{n, k+1}+ b_{n,\sigma (k+1)}\).