<i>ℐ</i> - <i>αβ</i> -statistical relative uniform convergence for double sequences and its applications

Abstract This article introduces a novel concept of convergence, referred to as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℐ</m:mi> </m:math> {\mathcal{ {\mathcal I} }} - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mi>β</m:mi> </m:math> \alpha \beta -statistical relative uniform convergence, for double sequences of functions. This notion, which is proposed for the first time in this article, is explored in depth, leading to the establishment of a Korovkin-type approximation theorem within this framework. The illustrative example demonstrates that the proposed convergence is indeed stronger than previously known forms. Additionally, this article investigates the rate of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi class="MJX-tex-caligraphic" mathvariant="script">ℐ</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msub> </m:math> {{\mathcal{ {\mathcal I} }}}_{2} - <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>α</m:mi> <m:mi>β</m:mi> </m:math> \alpha \beta -statistical relative uniform convergence, providing explicit computations to support the findings. The results contribute to the understanding of ideal statistical convergence and open up new perspectives for approximation theory.