Weighted means and summability by generalized Nörlund and other methods (Q1332236)

Work of the first author and \textit{T. Markovich} [J. Approximation Theory 68, No. 1, 49-55 (1992; Zbl 0745.40006)] is carried on, employing the generalized Nörlund matrix \((p_{n - k} p_ k/ \sum^ n_{k = 0} p_{n - k} p_ k : 0 \leq k \leq n)\) method \((N,p,p)\), as considered by the second author [Math. Z. 214, No. 2, 273-286 (1993; Zbl 0791.40004)]. Smoothness conditions on \(p_ n>0\) provided, allowing for \(p_ n = 1/n!\) to which the former results relate, limitability \((N,p,p)\) of \(s = (s_ n)\) is shown, at Theorem C, 1, e.g., to be implied by certain order conditions on some weighted means of \(s\), involving the order of \(1/\sqrt {g''}\), where \(g(n) = - \log p_ n\). The results apply to a scale of methodes \((N,p)_ \alpha\), say, such that \((N,p,p) = : (N,p)_ 1 \subseteq (N,p)_ 2 \subseteq \cdots \subseteq J_ p\), with the Abel- resp. Borel-type method \(J_ p\) (cf. the second quotation above).