On multivariate Lagrange interpolation

Abstract

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 formula for Lagrange interpolation of degree <italic>n</italic> of a function <italic>f</italic> , which is a sum of integrals of certain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n plus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(n + 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> st directional derivatives of <italic>f</italic> multiplied by simplex spline functions. We also provide two algorithms for the computation of Lagrange interpolants which use only addition, scalar multiplication, and point evaluation of polynomials.