RATES OF CONVERGENCE OF SEQUENCES OF KANTOROVICH OPERATORS NEAR $L^1[0,1]$ - Karim Munir Aszari, Abdul
This research examines the rate of convergence of the Kantorovich operator sequence in Lp[0, 1] spaces and grand Lebesgue spaces. The Kantorovich operator is a crucial tool in approximation theory used to approximate continuous functions. Within the context of Lp[0, 1] spaces, the convergence rate of the Kantorovich operator can be analyzed using the first and second derivatives of the approximated function. However, this analysis becomes more complex around L1[0, 1], motivating the use of grand Lebesgue spaces as a generalization of Lp[0, 1] spaces. grand Lebesgue spaces allow us to examine functions with complex and unbounded properties, which are challenging to analyze using traditional Lp[0, 1] approaches. In these spaces, the Kantorovich operator exhibits a more structured convergence rate, particularly when applied to functions with ?-H¨older continuity. These findings extend our understanding of the asymptotic properties of the Kantorovich operator and pave the way for further research in approximation theory and operator analysis. By employing grand Lebesgue spaces, this research provides new insights into the convergence behavior of the Kantorovich operator, which can be applied across various fields of mathematics and applied sciences. The results demonstrate that these spaces are highly effective in addressing cases where Lp[0, 1] spaces do not yield adequate results.