Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

Abstract

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. (Math. Methods Appl. Sci. 41 (2018), 671–683). The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

Keywords

• statistical convergence
• <i>P</i>-convergent
• statistically and relatively modular deferred-weighted summability
• relatively modular deferred-weighted statistical convergence
• Korovkin-type approximation theorem
• modular space
• convex space
• <i>N</i>-quasi convex modular
• <i>N</i>-quasi semi-convex modular