A unique multivariate Hermite interpolant on the simplex (Q1892562)

The author constructs a polynomial which interpolates given function values and derivatives at the vertices of a regular \(N\)-simplex. The interpolant is written as a linear combination of basis polynomials which are all generated from a basis polynomial defined on the standard simplex. The approximant interpolates quadratic polynomials exactly and has the lowest possible degree. Under these conditions, the interpolant is unique and the lowest possible degree turns out to be 3. Such interpolants are useful in optimization methods. There exist \(O(n^ 3)\) methods to evaluate function values, gradient and Hessian matrix at a given point.