COMPUTATIONAL ASPECTS OF POLYNOMIAL INTERPOLATION IN SEVERAL VARIABLES

The pair <THETA, P> of a point set THETA subset-of R(d) and a polynomial space P on R(d) is correct if the restriction map P --> R(THETA): p BAR-ARROW-POINTING-RIGHT p\THETA is invertible, i.e., if there is, for any f defined on THETA, a unique p is-an-element-of P which matches f on THETA. We discuss here a particular assignment THETA BAR-ARROW-POINTING-RIGHT PI(THETA), introduced by us previously, for which <THETA, PI(THETA)> is always correct, and provide an algorithm for the construction of a basis for PI(THETA), which is related to Gauss elimination applied to the Vandermonde matrix (curly-theta(alpha)) curly-theta is-an-element-of(THETA), alpha is-an-element-of Z(d)+ for THETA. We also discuss some attractive properties of the above assignment and algorithmic details, and present some bivariate examples.