Relative weighted almost convergence based on fractional-order difference operator in multivariate modular function spaces

Abstract

In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double difference operator $$\nabla ^{a,b,c}_h$$ . We first define the concepts of weighted almost $$\nabla $$ -statistical convergence and statistical weighted almost $$\nabla $$ -convergence of double sequences. By using the notion of relative uniform convergence involving a scale function $$\sigma $$ , we introduce modular relative weighted almost statistical convergence and modular relative statistical weighted almost convergence of double sequences. We then obtain some inclusion relations between these proposed methods and provide some counterexamples that show that these are non-trivial and proper extensions of the existing literature on this topic. Moreover, we apply the relative statistical weighted almost convergence of a double sequence of positive linear operators to prove some Korovkin-type theorems in multivariate modular spaces by considering several kinds of test functions. Finally, we present a non-trivial application to generalized Boolean sum (GBS) operators of bivariate generalized Bernstein–Durrmeyer operators.