Fuzzy measure-based statistical convergence and its applications in Korovkin-type approximation theory

This paper introduces a novel approach to statistical convergence by employing fuzzy measure theory, resulting in the concept of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> F -statistical convergence. Traditional methods of statistical and A -statistical convergence are extended through the development of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> F -density, which enables a more comprehensive analysis of convergence behavior in the context of fuzzy measures. A key contribution is a decomposition result, providing insight into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> F -statistical convergence by separating complex convergence behaviors into more manageable components. We also introduce triangular versions of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> F -statistical convergence, further adapting the theory to summability methods commonly used in approximation. By taking specific triangular fuzzy measure sequences, this framework reduces to A -statistical and classical statistical convergence, offering a unified perspective that broadens traditional convergence methods. Integrating summability theory with fuzzy measures, we derive new results in Korovkin-type approximation theory. We provide examples and theorems demonstrating that sequences of positive linear operators satisfy <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi mathvariant="script">F</m:mi> </m:math> F -statistical convergence, even under conditions that do not meet the criteria of classical and statistical Korovkin theorems. This work not only generalizes existing theories but also opens new avenues for applying fuzzy statistical convergence in operator theory and functional analysis.