Summability Matrices That Preserve Asymptotic Equivalence for Ideal Convergence
DOI:
https://doi.org/10.5644/SJM.12.1.08Keywords:
Ideal convergence, statistical convergence, asymptotic equivalence, statistical limit superiorAbstract
A characterization of summability matrices that preserve the asymptotic equivalence of two sequences is given, where asymptotic
equivalence is defined with respect to an ideal $I$ of subsets of $\mathbb{N}$. Pobyvanets (1980) gave necessary and sufficient conditions for a nonnegative summability matrix $A$ to have the property that, for two nonnegative sequences $x$ and $y$ bounded away from 0, the ratio $Ax/Ay$ tends to 1 whenever $x/y$ tends to 1; an analogous characterization is given where the convergence of $x/y$ is with respect to an ideal $I$ as well as a new proof of Pobyvanets’ theorem. Similar extensions are given of theorems
of Marouf and Li, in particular characterizations of summabilty matrices that map sequences $x$ and $y$ such that $x/y$ is $I$-convergent to 1 to sequences with the property that the ratio of the sums of their tails or the ratio of the supremum of their tails also tend to 1 with respect to an ideal J.
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