On multivariate Lagrange interpolation

On Multivariate Lagrange Interpolation

Lagrange interpolation by polynomials in several variables is studied through a finite difference approach. We establish an interpolation formula analogous to that of Newton and a remainder formula, both of them in terms of finite differences. We prove that the finite difference admits an integral representation involving simplex spline functions. In particular, this provides a remainder formul...

A continuity property of multivariate Lagrange interpolation

Let {St} be a sequence of interpolation schemes in Rn of degree d (i.e. for each St one has unique interpolation by a polynomial of total degree ≤ d) and total order ≤ l. Suppose that the points of St tend to 0 ∈ Rn as t→ ∞ and the Lagrange-Hermite interpolants, HSt , satisfy limt→∞HSt(x) = 0 for all monomials xα with |α| = d + 1. Theorem: limt→∞HSt (f) = T d(f) for all functions f of class Cl−...

Intertwining unisolvent arrays for multivariate Lagrange interpolation

Let Pd(C ) denote the space of polynomials of degree at most d in n complex variables. A subset X of C – we will usually speak of configuration or array – is said to be unisolvent for Pd(C ) (or simply unisolvent of degree d) if, for every function f defined on X there exists a unique polynomial P ∈ Pd(C ) such that P(x) = f (x) for every x ∈ X. This polynomial is called the Lagrange interpolat...

On Lagrange multivariate interpolation problem in generalized degree polynomial spaces

The aim of this paper is to study the Lagrange multivariate interpolation problems in the space of polynomials of w-degree n. Some new results concerning the polynomial spaces of w-degree n are given. An algorithm for obtaining the w-minimal interpolation space is presented. Key–Words: Lagrange multivariate polynomial interpolation, whomogeneous polynomial spaces, Generalize degree.

On the Average Errors of Multivariate Lagrange Interpolation