A new class of differential operators on the simplex is introduced, which define weighted Sobolev norms andwhose eigenfunctions are orthogonal polynomialswith respect to Jacobiweights. These operators appear naturally in the study of quasi-interpolants which are intermediate between Bernstein– Durrmeyer operators and orthogonal projections on polynomial subspaces. The quasi-interpolants satisfy...

In this paper we present two new alternative ways for the proof of Voronovskaja-type formula of the Bleimann, Butzer and Hahn bivariate operators, using the close connection between the recalled operators and Bernstein bivariate operators, respectively Stancu bivariate operators.

In this note, we give an elaboration of a basic problem on convergence theorem of (p,q)-analogue of Bernstein-type operators. By some classical analysis techniques, we derive an exact class of (pn,qn)-integer satisfying limn→∞[n]pn ,qn =∞ with limn→∞ pn = 1 and limn→∞ qn = 1 under 0 < qn < pn ≤ 1. Our results provide an erratum to corresponding results on (p,q)-analogue of Bernstein-type operat...