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