$\mathcal{F-}$relative $\mathcal{A-}$summation process for double sequences and abstract Korovkin type theorems

Authors : Sevda YILDIZ

Pages : 1047-1062

Doi:10.15672/hujms.796762

View : 14 | Download : 3

Publication Date : 2021-08-06

Article Type : Research Paper

Abstract :In this paper, we first introduce the notions of $\mathcal{F-}$relative modular convergence and $\mathcal{F-}$relative strong convergence for double sequences of functions. Then we prove some Korovkin-type approximation theorems via $\mathcal{F-}$relative $\mathcal{A}-$summation process on modular spaces for double sequences of positive linear operators. Also, we present a non-trivial application such that our Korovkin-type approximation results in modular spaces are stronger than the classical ones and we present some estimates of rates of convergence for abstract Korovkin-type theorems. Furthermore, we relax the positivity condition of linear operators in the Korovkin theorems and study an extension to non-positive operators.
Keywords : abstract Korovkin theorem, double sequence, filter convergence, matrix summability, modular spaces