Some properties of bivariate Bernstein-Schoenberg operators

Abstract

The bivariate Bernstein-Schoenberg operator Vr of degree m, introduced in [5], is a spline approximation operator that generalizes the Bernstein polynomial operator B,~. It is shown here that for a convex function f, f<-vr(f) <-Bin(f). This result is then used to show that far a twice differentiable function g, the asymptotic error lim m(Vr(g)-g) depends only on the asymptotic error for quadratic polynomials. The latter is evaluated explicitly in the special circumstances that V r is, in a sense, asymptotically close to B,,.

A Definition of Two-Dimensional Schoenberg Type Operators

“…The subject of multivariate splines was approached by different methods and from various points of view, such as in the papers written by: Curry and Schoenberg [6], Goodman and Lee [7], de Boor and Hollig [8], Karlin et al [9], Goodman [10], Chui [11], Schumaker [12], Conti and Morandi [13], Ugarte et al [14], and Groselj and Knez [15]. Curry and Schoenberg indicated in [6] that the multivariate spline functions can be constructed from volumes of slices of polyhedra; therefore, papers can be found that were written toward that direction.…”

“…For example, this idea led to the recurrence relations for multivariate splines presented by Karlin et al in [9]. Goodman and Lee in [7] and Goodman in [10] approached the subject of multidimensional Bernstein-Schoenberg operators depending on m-dimensional volume. In [8] the subject of multidimensional B-splines is treated by de Boor and Hollig as the m-shadow of the polyhedral convex body included in R n .…”