On the necessary conditions for absolute weighted arithmetic mean summability factors (Q1263752)

Let \(\sum a_ n\) be a given infinite series with the sequences of partial sums \((s_ n)\) and let \((p_ n)\) be a sequence of positive numbers such that \(P_ n=\sum^{n}_{\nu =0}p_{\nu}\to \infty\) as \(n\to \infty\) \((P_{-i}=p_{-i}=0\), \(i\geq 1)\). The sequence-to-sequence transformation \(t_ n=\frac{1}{P_ n}\sum^{n}_{\nu =0}p_{\nu}s_{\nu}\) defines the sequence \((t_ n)\) of the \((\bar N,p_ n)\) mean of the sequence \((s_ n)\), generated by the sequence of coefficients \((p_ n)\). The series \(\sum a_ n\) is said to be summable \(| \bar N,p_ n|_ k\), \(k\geq 1\), if \(\sum^{\infty}_{n=1}(P_ n/p_ n)^{k-1}| t_ n-t_{n- 1}|^ k<\infty;\) it is said to be bounded \([\bar N,p_ n]_ k\), \(k\geq 1\), if \(\sum^{n}_{\nu =1}p_{\nu}| s_{\nu}|^ k=O(P_ n)\) as \(n\to \infty\) [see the first author, Tamkang J. Math. 16, 13-20 (1985; Zbl 0592.40006)]. In this paper the following theorem is proved. Theorem. Let \(\sum a_ n\) be bounded \([\bar N,p_ n]_ k\). If \(\sum a_ n\lambda_ n\) is summable \(| \bar N,p_ n|_ k\), then the following conditions are necessary: \(\lambda_ n=O(1)\) and \(\Delta \lambda_ n=(P_ n/p_ n)^{1/k}\) as \(n\to \infty\), \(k\geq 1\).