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### Multivariate polynomial interpolation: The least interpolant

Here is a commented list of relevant papers. Complete references can be found in the spline bibliography), under the handle indicated (handles start with a percent sign).

The first paper on the least interpolant is On multivariate polynomial interpolation (%BoorRon90) but I would read first the shorter, lighter and much later paper On the error in multivariate polynomial interpolation (%Boor92). An early (incomplete) overview is given in Polynomial interpolation in several variables (%Boor94b)

For the calculation of the interpolant, see Computational aspects of polynomial interpolation in several variables (%BoorRon92b). The connection of the calculation to Gauss elimination by segments is discussed in Gauss elimination by segments and multivariate polynomial interpolation (%Boor94a). M-files, i.e., subroutines in MATLAB, for the construction and evaluation of least interpolants in any number of variables can be found in the list of m-files .

The extension, from matching function values to arbitrary data, i.e., from the data (g(theta): theta in Theta) to the data (lambda(g) : lambda in Lambda), for an arbitrary finite collection Lambda of linear functionals on the multivariate polynomials, is carried out in The least solution for the polynomial interpolation problem (%BoorRon92a). One view on multivariate Hermite interpolation is expressed in the following talk from 13apr99. But, as of dec06, my view is different; see, e.g., What are the limits of Lagrange projectors?

A first step toward an error formula for the least interpolant is taken in A multivariate divided difference (%Boor95a), and is used in On the Sauer-Xu formula for the error in multivariate polynomial interpolation (%Boor96a) to rederive one result of Sauer and Xu in ``On multivariate Lagrange interpolation'', Math.Comp.; 64; 1995; 1147--1170; MR 95j:41041 (%SauerXu95a). The details missing in %Boor95 are supplied in The error in polynomial tensor-product, and in Chung-Yao, interpolation (%Boor97a).