Best bounds on the approximation of polynomials and splines by their control structure

Abstract

We present best bounds on the deviation between univariate polynomials, tensor product polynomials, Bézier triangles, univariate splines, and tensor product splines and the corresponding control polygons and nets. Both pointwise estimates and bounds on the L p -norm are given in terms of the maximum of second differences of the control points. The given estimates are sharp for control points corresponding to arbitrary quadratic polynomials in the univariate case, and to special quadratic polynomials in the bivariate case.