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The concept of equi-statistical convergence is more general than that of the well-established statistical uniform convergence. In this paper, we have introduced the idea of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence under the difference operator including $(p, q)$-integers via deferred Nörlund statistical convergence so as to build up a few inclusion relations between them. We have likewise presented the notion of the deferred weighted (Nörlund type) equi-statistical convergence (presumably new) in view of difference sequence of order $r$ based on $(p, q)$-integers to demonstrate a Korovkin type approximation theorem and proved that our theorem is a generalization (non-trivial) of some well-established Korovkin type approximation theorems which were demonstrated by earlier authors. Eventually, we set up various fascinating examples in connection with our definitions and results.
}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2018-0018}, url = {http://global-sci.org/intro/article_detail/ata/23861.html} }The concept of equi-statistical convergence is more general than that of the well-established statistical uniform convergence. In this paper, we have introduced the idea of equi-statistical convergence, statistical point-wise convergence and statistical uniform convergence under the difference operator including $(p, q)$-integers via deferred Nörlund statistical convergence so as to build up a few inclusion relations between them. We have likewise presented the notion of the deferred weighted (Nörlund type) equi-statistical convergence (presumably new) in view of difference sequence of order $r$ based on $(p, q)$-integers to demonstrate a Korovkin type approximation theorem and proved that our theorem is a generalization (non-trivial) of some well-established Korovkin type approximation theorems which were demonstrated by earlier authors. Eventually, we set up various fascinating examples in connection with our definitions and results.