# zbMATH — the first resource for mathematics

Asymptotic equivalence and summability. (English) Zbl 0788.40001
This paper studies the relationships between the asymptotic equivalence of two sequences $$(\lim_ n x_ n/y_ n=1$$; i.e. $$x\sim y)$$ and three variations for this equivalence. Let $$A$$ be a nonnegative sequence-to- sequence transformation matrix, $$R_ m Az:= \sum_{n\geq m} |(Az)_ n|$$, $$S_ m Az:= \sum_{n\leq m} | (Az)_ n|$$, and $$\mu_ m Az:=\sup_{n\geq m} | (Az)_ n|$$. The three variations are given by $$R_ m Ax/R_ m Ay$$, $$S_ m Ax/S_ m Ay$$, and $$\mu_ m Ax/\mu_ m Ay$$. A typical result is the following. If $$x$$ and $$y$$ are nonvanishing null sequences such that $$x\sim y$$, then $$\mu x\sim\mu y$$. A study of these relationships when the limit is zero was done by J. A. Fridy [Can. J. Math. 30, 808-816 (1978; Zbl 0359.40003)].

##### MSC:
 40C05 Matrix methods for summability
##### Keywords:
weighted mean matrices; asymptotic equivalence
Full Text: