Constructive Approximation

, Volume 6, Issue 3, pp 287–302

On multivariate polynomial interpolation

  • Carl de Boor
  • Amos Ron

DOI: 10.1007/BF01890412

Cite this article as:
de Boor, C. & Ron, A. Constr. Approx (1990) 6: 287. doi:10.1007/BF01890412


We provide a map
which associates each finite set Θ in complexs-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 spacesQ from which interpolation at Θ is uniquely possible, our πΘ 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 PΘ, and, for given smoothf, a polynomialqQ is sought for which
$$p(D)(f - q)(\theta ) = 0, \forall p \in P_\theta , \theta \in \Theta $$

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 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.

Key words and phrases

ExponentialsPolynomialsMultivariateInterpolationNewton formBirkhoff interpolation

Copyright information

© Springer-Verlag New York Inc. 1990

Authors and Affiliations

  • Carl de Boor
    • 1
  • Amos Ron
    • 1
  1. 1.Center for Mathematical SciencesUniversity of Wisconsin-MadisonMadison WisconsinUSA