A Sequence of Kantorovich-Type Operators on Mobile Intervals

Abstract

In this paper, we introduce and study a new sequence of positive linear operators, acting on bothspaces of continuous functions as well as spaces of integrable functions on [0, 1]. We state some qualitative propertiesof this sequence and we prove that it is an approximation process both in C([0, 1]) and in Lp([0, 1]), also providingsome estimates of the rate of convergence. Moreover, we determine an asymptotic formula and, as an application,we prove that certain iterates of the operators converge, both in C([0, 1]) and, in some cases, in Lp([0, 1]), to a limitsemigroup. Finally, we show that our operators, under suitable hypotheses, perform better than other existing ones inthe literature.