Multivariate weighted kantorovich operators

Abstract

Recently, some weighted Durrmeyer type operators were used as research tools in Learning Theory. In this paper we introduce a class of multidimensional weighted Kantorovich operators Kn on C(Q d ) where Q d is the d-dimensional hypercube [0, 1] d . We show that each Kn has a unique invariant probability measure and determine this measure. Then, using results from approximation theory and the theory of ergodic operators, we find the limit of the iterates of Kn and give rates of convergence of the iterates toward the limit. Finally, we show that some Kantorovich type operators previously investigated in literature fall into the class of operators introduced in our paper. Other properties and applications, involving Learning Theory, will be presented in a forthcoming paper, where we will consider also operators on spaces of Lebesgue integrable functions on the hypercube Q d .