ON MULTIVARIATE POLYNOMIAL INTERPOLATION

We provide a map which associates each finite set Θ in complex s-space with a polynomial space πΘ from which interpolation to arbitrary data given at the points in Θ is possible and uniquely so. Among all polynomial spaces Q from which interpolation at Θ is uniquely possible, our πΘ is of smallest degree. It is also D- 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 each gq∈Θ, there is associated a polynomial space PΘ, and, for given smooth f, a polynomial q∈Q is sought for which {Mathematical expression}. We obtain πΘ as the "scaled limit at the origin" (expΘ)↓ of the exponential space expΘ with frequencies Θ, and base our results on a study of the map H→H↓ defined on subspaces H of the space of functions analytic at the origin. This study also allows us to determine the local approximation order from such H and provides an algorithm for the construction of H↓ from any basis for H. © 1990 Springer-Verlag New York Inc.