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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
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