Abstract
It is known that given a regular matrix A and a bounded sequence x there is a subsequence (respectively, rearrangement, stretching) y of x such that the set of limit points of Ay includes the set of limit points of x. Using the notion of a statistical limit point, we establish statistical convergence analogues to these results by proving that every complex number sequence x has a subsequence (respectively, rearrangement, stretching) y such that every limit point of x is a statistical limit point of y. We then extend our results to the more general A‐statistical convergence, in which A is an arbitrary nonnegative matrix.
Highlights
In [2, 3] Buck characterized convergence by proving that if x is a nonconvergent sequence, no regular matrix can sum every subsequence of x. This result was extended by Agnew [1] who showed that given a regular matrix A and a bounded sequence x, there is a subsequence y of x such that the set of limit points of Ay includes the set of limit points of x
To see that λj is a statistical limit point of y, we show that w is a nonthin subsequence of y
We show that a sequence x has a rearrangement z such that every limit point of x is a statistical limit point of z
Summary
In [2, 3] Buck characterized convergence by proving that if x is a nonconvergent sequence, no regular matrix can sum every subsequence of x. If x is a complex number sequence with a countably infinite set of (finite) limit points D = {λj}∞j=1, there is a rearrangement z of x such that every λj in D is a statistical limit point of z.
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