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Asymptotic equivalence of sequences and summability. (English) Zbl 0895.40001

For a bounded real-valued sequence \(x=(x_n)\) let \(\mu_mx= \sup_{n\geq m} | x_n|\), \(m=0, 1, 2 \dots\). Two sequences \(x\) and \(y\) are called asymptotically equivalent \((x\approx y)\) if \(\lim_n(x_n/y_n)=1\). It is known [see I. P. Pobyvanets, Mat. Fiz. 28, 83-87 (1980; Zbl 0447.40005)] that a nonnegative infinite matrix \(A=(a_{ki})\) preserves the asymptotic equivalence of nonnegative sequences if and only if (*) \(\lim_n (a_{ni}/ \sum_{k\geq 0} a_{nk}) =0\) for each \(i\). The author proves the following results, where \(x\) and \(y\) are sequences such that \(x_n\), \(y_n> \delta\) \((n=0, 1, \dots)\) for some \(\delta>0\). (1) \(x\approx y\) implies \(\mu Ax\approx \mu Ay\) if and only if \(A\) satisfies the condition (*). (2) \(\mu x\approx \mu y\) implies \(\mu Ax \approx \mu Ay\) if and only if \[ \text{the sequence } (\sum_{j\geq 0} a_{kj})_{k\geq 0} \text{ is bounded}, \tag{i} \]
\[ \lim_n (\sup_{k>n} a_{kj})/(\sup_{k\geq n} \sum_{i\geq 0} a_{ki})= 0\text{ for each }j, \tag{ii} \]
\[ \lim_n (\sup_{k>n} \sum_{i\geq 1} a_{kj_i})/(\sup_{k\geq n} \sum_{i\geq 0}a_{ki}) =1\text{ for any infinite sequence } j_1 <j_2< \dots. tag iii \] {}.
Reviewer: T.Leiger (Tartu)

MSC:

40C05 Matrix methods for summability

Citations:

Zbl 0447.40005
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