On interpolation by rational functions with prescribed poles with applications to multivariate interpolation

摘要

This paper is concerned with interpolation in the sense of Hermite by certain rational functions of one or several complex variables. In the univariate setting the interpolants are generalized polynomials of a Cauchy—Vandermonde space, whereas in the multivariate setting the interpolants are elements of suitable subspaces of tensor products of Cauchy—Vandermonde spaces.A Newton-type algorithm is given computing an interpolating univariate rational function with prescribed poles with no more than O(M2) arithmetical operations where M is the number of nodes. It is proved that the generalized divided differences are analytic functions of the nodes if the function to be interpolated is analytic. The algorithm will be extended to the multivariate setting. For subsets of grids possessing the rectangular property and for certain subspaces of a tensor product of Cauchy—Vandermonde spaces an algorithm computing an interpolating rational function of two variables is given whose complexity is O(m2n+n2m), where m and n are the numbers of interpolation points in the x- and y-direction, respectively.