Rate of convergence of Kantorovich operator sequences near L1([0, 1])

Abstract

The study of the rate of convergence of Kantorovich operator sequences has predominantly focused on the Lp spaces for 1< p<infty, yet the behaviour near L1([0,1]) remains less understood, particularly as p approaches 1. To bridge this gap, we investigate the rate of convergence within the framework of the grand Lebesgue spaces Lp ([0,1]), which encompass all Lp ([0,1]) spaces for 1<p<infty but remain a subset of L1([0,1]).

Our approach leverages the intrinsic properties of Lp ([0,1]) to derive new results on the convergence rate of Kantorovich operator sequences. Specifically, our objective is to demonstrate that Kantorovich operators exhibit a significant rate of convergence within this broader context, thereby providing insights applicable to the boundary behavior as p to 1.

We will then apply these findings to alpha-Holder continuous functions to further understand the convergence rate of Kantorovich operator sequences in these settings. This combined approach suggests that functions with derivatives in Lp ([0,1]) exhibit specific convergence rates under Kantorovich operators.