language: en

On multivariate polynomial interpolation

Carl de Boor 1 & Amos Ron

ABSTRACT We provide a map fi 7! fi which associates each finite set fi of points in C s with a polynomial space fi from which interpolation to arbitrary data given at the points in fi is possible and uniquely so. Among all polynomial spaces Q from which interpolation at fi is uniquely possible, our fi 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 2 fi, there is associated a polynomial space P , and, for given smooth f , a polynomial q 2 Q is sought for which p(D)(f q)() = 0; all p 2 P ; 2 fi: We obtain fi as the "scaled limit at the origin" (Exp fi ) # of the exponential space Exp fi with frequencies fi, and base our results on a study of the map H 7! 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.

AMS (MOS) Subject Classifications: primary 41A05, 41A63, 41A10; secondary 65D05, 41A30 Key Words: exponentials, polynomials, multivariate, interpolation, Newton form, Birkhoff interpolation

[Authors' affiliation and address:]

Center for Mathematical Sciences University of Wisconsin-Madison
610 Walnut St. Madison WI 53705

[1 supported in part by the National Science Foundation under Grant No. DMS-8701275 and by the United States Army under Contract No. DAAL03-87-K-0030]