Two formulas for the general multivariate polynomial which interpolates a regular grid on a simplex

Two formulas are exhibited for the multivariate Lagrange shape polynomials which interpolate a regular grid on a simplex in R". 1. Introduction. The idea of a shape function (or Lagrange) basis for an approxi- mating function subspace is fundamental to the finite element method, spline theory and interpolation procedures in general. Given a set of points (or nodes) in the domain of the approximants, a shape function is an approximant associated with a given node and which assumes the value 1 at that node and the value zero at all other nodes in the set. If we have a shape function for each node, we say that the set of shape functions is biorthonormal to the node set. It is then very natural to attempt to force an approximating function to interpolate the node set (i.e., to take on arbitrary values at the given points) by constructing it as a linear combination of the shape functions. The coefficients in the linear combination are the arbitrary values to be assumed and the biorthonormal property ensures, in an obvious and elemen- tary way, that this linear combination will indeed interpolate those values at the given nodes. Less obvious, but equally elementary, is the fact that biorthonormality of the shape functions implies that they are unique and linearly independent and so form a basis for the approximating subspace. To state all this precisely, let I tea vector space over an ordered field F of character zero, and let !F be a family of functions defined on X and with values in F, which is itself a vector space over F by means of the operations induced from F. Let ?F be an (approximating) subspace of J^ of finite dimension TV and let the node set P = {x{,)}"_x be a subset of X. Using straightforward arguments of elementary linear algebra, as is done by Davis in Chapter 2 of (1) and Thacher (5), (6), it is easy to prove the following