The discretization for bivariate ideal interpolation

Abstract

a b s t r a c tCarl de Boor conjectured that every ideal interpolant over complex field is the pointwise limit of Lagrange interpolants. Boris Shekhtman proved that the conjecture is true in two variables, and he also provided a counterexample for more than three variables. However, Shekhtman only mentioned the existence of some mathematical objects (without giving a method to compute them) in his proof of the bivariate case. For general interpolation condition functionals on the interpolation sites, we improve Shekhtman's method to find a sequence of interpolation sites (also called discrete sites), such that the corresponding Lagrange interpolants converge to the given bivariate ideal interpolant. We discuss a special case where the multiplicity space is of breadth one. The results in this paper give a completely algorithmic way to realize Shekhtman's method.

Computing Groebner bases of ideal interpolation

“…Applying Taylor expansions to the original algorithm, Farr and Gao's method can be applied to compute the vanishing ideal when the interpolation points have multiplicities, but the multiplicity set needs to be a delta set in N d . To avoid solving linear equations, Lederer [7] gives an algorithm to compute the Gröbner basis of an arbitrary finite set of points under lexicographic order by induction over the dimension d. Jiang, Zhang and Shang [8] give an algorithm for computing the Gröbner basis of a single point ideal interpolation. All the above methods are considered for special cases.…”