Reference Functional and Characteristic Space for Lagrange and Bernstein Operators

S. De Marchi and M. Morandi Cecchi

Approximation Theory & Appl. Vol. 11 (4), pp. 6--14 1995.


Abstract

This paper deals with the description and the representation of polynomials defined over $n$-simplices. The polynomials are computed by using two recurrent schemes: the {\em Neville-Aitken} one for the Lagrange interpolating operator and the {\em De Casteljau} one for the Bernstein-B\'ezier approximating operator. Both schemes fall into the framework of transformations of the form $ {\cal F}_i^{i+p}(x)=\sum_{j=i}^{i+p} c_{i,j}^p(x) F_j$ where the F_j are given numbers (for example, at the initial step they coincide with the values of the function on a given lattice), and the coefficients $c_{i,j}^p(x)$ are linear polynomials valued in $x$. A general theory for such sequence of transformations can be found in \cite{kn:1} where it is also proved that these tranformations are completely characterized in term of a linear functional: reference functional. This functional is associated with a linear space: characteristic space. \\ The concepts of reference functionals and characteristic spaces will be used and we shall prove the existence of a characteristic space for the reference functionals associated with these operators.

Keywords: Neville-Aitken's interpolating form, Bernstein-B\'ezier's approximating form, lattice, reference functional, characteristic space.