On multivariate polynomial interpolation

Abstract

We provide a map

which associates each finite set Θ in complexs-space with a polynomial space π_Θ from which interpolation to arbitrary data given at the points in Θ is possible and uniquely so. Among all polynomial spacesQ from which interpolation at Θ is uniquely possible, our π_Θ is of smallest degree. It is alsoD- and scale-invariant. Our map is monotone, thus providing a Newton form for the resulting interpolant. Our map is also continuous within reason, allowing us to interpret certain cases of coalescence as Hermite interpolation. In fact, our map can be extended to the case where, with eachgq∈Θ, there is associated a polynomial space P_Θ, and, for given smoothf, a polynomialq∈Q is sought for which

$$p(D)(f - q)(\theta ) = 0, \forall p \in P_\theta , \theta \in \Theta $$

.

We obtain π_Θ as the “scaled limit at the origin” (exp_Θ)↓ of the exponential space exp_Θ with frequencies Θ, and base our results on a study of the mapH→H _↓ defined on subspacesH of the space of functions analytic at the origin. This study also allows us to determine the local approximation order from suchH and provides an algorithm for the construction ofH _↓ from any basis forH.