New polynomial preserving operators on simplices: direct results

Abstract

A new class of differential operators on the simplex is introduced, which define weighted Sobolev norms and whose eigenfunctions are orthogonal polynomials with respect to Jacobi weights. 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 a Voronovskaja-type identity and a Jackson–Favard-type error estimate. These and further properties follow from a spectral analysis of the differential operators. The results are based on a pointwise orthogonality relation of Bernstein polynomials that was recently discovered by the authors.