-------------------------------------> last change: 07jun06
### 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).