A class of weighted means as potent conservative methods. (English) Zbl 0880.40003

Let \(\chi\) be the set of all sequences of 0’s and 1’s. For an infinite matrix \(H=(h_{nk})\) denote \((H)\) the class of all \(H\)-summable bounded sequences. The authors study the following subclasses of the class \(K\) of real conservative matrices \(A=(a_{nk})\): \[ \begin{aligned} KF &=\{A\in K\mid A\text{ sums all almost convergent sequences}\}, \\ KL &=\{A \in K\mid \sup_k |a_{nk} -\lim_na_{mk} |\to 0\;(n \to\infty) \}, \\ KG &=\{A \in K\mid H\in K\text{ for every matrix } H\text{ such that }(H)\supset(A) \cap\chi \}, \\ KM &=\{A \in K\mid (H)\supset (A)\text{ for every matrix } H\in K \text{ such that } (H) \supset (A) \cap\chi \}, \\ KP &=\{A\in K\mid H\in K \text{ and } (H) \supset (A) \text{ for every matrix } H\text{ such that } (H) \supset (A) \cap\chi \}=\\ KG & \cap KM. \end{aligned} \] The matrices \(A\in KP\) are called potent. In [Bull. Lond. Math. Soc. 26, No. 3, 297-302 (1994; Zbl 0812.40003)] the authors have proved that \(A\in KF \Leftrightarrow A\in KL \Leftrightarrow A\in KG \Leftrightarrow A\in KM \Leftrightarrow A\in KP \Leftrightarrow \lim_n a_{nm} =0\) for any conservative Hausdorff matrix \(A\). In the present paper they consider regular positive weighted mean methods \(A= (\overline N, p_n)\) defined by \[ y_n= {1\over P_n} \sum^n_{k= 0} p_k x_k \quad (n\in\mathbb{N}), \text{ where } P_n= \sum^n_{k=0} p_k \quad (p_k>0 \text{ and } P_n \to\infty), \] and prove that the following statements are equivalent: (a) \(A\in KG\), (b) \(A\in KL\), (c) \(p_n= o(P_n)\), (d) \(A\in KP\). It is also shown that (i) (c) implies \(A\in KM\), but the converse implication is not true, and (ii) there exists \(A= (\overline N,p_n)\in KP\) such that \(A \notin KF\).
Reviewer: T.Leiger (Tartu)


40D20 Summability and bounded fields of methods
40C05 Matrix methods for summability
40G05 Cesàro, Euler, Nörlund and Hausdorff methods


Zbl 0812.40003