Abstract
We introduce the notion of statistical weighted A-summability of a sequence and establish its relation with weighted A-statistical convergence. We also define weighted regular matrix and obtain necessary and sufficient conditions for the matrix A to be weighted regular. As an application, we prove the Korovkin type approximation theorem through statistical weighted A-summability and using the BBH operator to construct an illustrative example in support of our result.
Highlights
Introduction and preliminariesThe term ‘statistical convergence’ was first presented by Fast [ ]; it is a generalization of the concept of ordinary convergence
A root of the notion of statistical convergence can be detected by Zygmund [ ], where he used the term ‘almost convergence’ which turned out to be equivalent to the concept of statistical convergence
2 Statistical weighted A-summability In the present section we introduce the notion of statistical weighted A-summability and prove that this method of summability is stronger than the weighted A-statistically convergent notion
Summary
= , we say that s = (sk)k∈N is weighted summable, called cN -summable, to some number L. We shall write cN -lim s = L and cNdenotes the space of all weighted summable sequences. A sequence s = (sk)k∈N is said to be weighted A-summable if the Atransform of s is weighted summable. It is said to be weighted A-summable to L if the A-transform of s is weighted summable to L, that is, m∞. We prove the following characterization of a weighted regular matrix. = L = lim sk (since was arbitrary) This shows that A is weighted regular. ), the A-transforms of the sequence ek and e belong to cNand ek ∈ c gives condition ( ) and e ∈ c proves the validity of ( )
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