## 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)

### MSC:

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

Zbl 0812.40003