Revisiting Korovkin-type Theorems in Banach Function Spaces

Abstract: This article delves into Korovkin-type theorems in Banach function spaces, as established by Yusuf Zeren et al. (2022). We prove that in this theorem, the positivity of the operators is not a necessary requirement and provide example of a non positive operator where it is applicable. Under the assumption of positivity, we establish an operator version of the result. Additionally, we derive a quantitative form of the result using the modulus of continuity. We apply the result to examples such as Lebesgue space, Weighted Lebesgue space, Grand Lebesgue space, etc. Furthermore, we present numerical illustrations for specific cases.