## On absolute summability.(English)Zbl 0649.40009

Let $$\sum^{\infty}_{n=0}x_ n$$ be an infinite series with partial sums $$s_ n$$. The series $$\sum^{\infty}_{n=0}x_ n$$ is called absolutely summable (C,k) with index p, or simple summable $$| C,k| p$$, $$p\geq 1$$, if $$\sum^{\infty}_{n=1}n^{-1}| \tau^{k}_{n}| \quad p<\infty,\tau^{k}_{n}$$ being the nth Cesàro mean of order k $$(k>-1)$$ of the sequence $$(nx_ n)$$. Let $$A=(a_{nk})$$ be a normal matrix i.e. lower semi matrix with non-zero diagonal entries. By $$(T_ n)$$ we denote the A-transform of a sequence $$(s_ n)$$, where $$T_ n=\sum^{\infty}_{v=0}a_{nv}s_ v=\sum^{n}_{v=0}a_{nv}s_ v.$$ A series $$\sum^{\infty}_{n=0}x_ n$$ is called summable $$| A| p$$, $$p\geq 1$$, if $$\sum^{\infty}_{n=1}n^{p-1}\cdot | T_ n-T_{n- 1}| \quad p<\infty.$$ The author establishes a relation between the summability methods $$| C,1| p$$ and $$| A| p$$, $$p\geq 1$$, details of which are cited in the paper.
Reviewer: B.P.Mishra

### MSC:

 40F05 Absolute and strong summability

### Keywords:

Cesàro mean; A-transform