On a generalization of Kantorovich operators on simplices and hypercubes

Abstract

In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the N-dimensional hypercube and on the N-dimensional simplex (N ≥ 1). These operators represent a natural generalization to the multidimensional setting of the ones introduced in [Altomare and Leonessa, Mediterr. J. Math. 3: 363–382, 2006] and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the Lp-norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.