Some remarks on absolute summability methods.(English)Zbl 0879.40005

Let $$\sum^{\infty}_{n=0} a_n$$ be a given infinite series with partial sums $$s_n$$, $$n=0,1,\dots$$, and $$(C,\alpha)$$ means $$\sigma^{\alpha}_n$$, $$\alpha >-1$$, $$n=0,1,\ldots$$. In what follows, we suppose $$p_k >0$$, $$k=0,1,\dots$$, $$\sum^\infty_{k=0} p_k = \infty$$, and set $$P_n = \sum^n_{k=0} p_k$$ and $$T_n = P_n^{-1} \sum^n_{k=0} p_k s_k$$, $$n = 0,1,\dots$$. The series $$\sum^{\infty}_{n=0} a_n$$ is said to be summable $$|C,\alpha |_k$$, $$\alpha >-1$$, $$k\geq 1$$, if $$\sum^{\infty}_{n=1} n^{k-1} \cdot |\Delta \sigma_{n-1}^{\alpha}|^k < \infty$$. It is said to be summable $$|\overline {N}, p_n|_k$$, $$k \geq 1$$ if $$\sum^{\infty}_{n=1}(P_n/p_n)^{k-1} \cdot |\Delta T_{n-1}|^k < \infty$$.
Let us assume now that $$q_n >0$$, $$n = 0,1,\dots$$ and $$Q_n = \sum^n_{j=0} q_j \to \infty$$ as $$n\to \infty$$. H. Bor and B. Thorpe [Analysis 7, 145-152 (1987; Zbl 0639.40005)] showed that if for some $$A>0$$ and $$B>0$$ $A \cdot \frac {q_n}{Q_n} \leq \frac {p_n}{P_n} \leq B \cdot \frac {q_n}{Q_n}, \;n = 0,1,\dots,\tag{1}$ then $$|\overline {N},p_n|_k \Leftrightarrow |\overline {N},q_n|_k, \;k \geq 1$$. (The symbol $$\Leftrightarrow$$ is used here to denote the equivalence of two summability methods.)
The author of this paper presents (in Section 1) a detailed detection of this assertion in the particular case $$q_n =1$$, $$n=0,1,\dots$$, i.e., for $$|C,1|_k$$ summability. Further, he points out (in Section 2) that the condition (1) in the Bor-Thorpe assertion mentioned above can be somewhat relaxed. Finally, a little comment on a similar result of Bor on $$|C,\alpha |_k$$ summability, $$0<\alpha \leq 1$$, $$k\geq 1$$ is added.
All propositions of the paper are easy to prove using well-known techniques from calculus.

MSC:

 40F05 Absolute and strong summability 40D25 Inclusion and equivalence theorems in summability theory 40G05 Cesàro, Euler, Nörlund and Hausdorff methods

Zbl 0639.40005