On Multivariate Lagrange Interpolation

Abstract

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 « of a function /, which is a sum of integrals of certain (n + l)st directional derivatives of / 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.

Approximation of boundary element matrices

“…It gives an upper bound for the error of multivariate polynomial interpolation. We will use the same notations as in [18]. …”

Study on Time Synchronization of Marine Fault Recording Device Based on Lagrange Once Interpolation Algorithm

“…1 that Lagrange interpolation is related to the number of original sampling points involved in the operation [6] , and the following takes the current function i(t) as an example to analyze Lagrange interpolation. Suppose that the host receives two packets sent by the same merging unit and obtains two discrete points [tk,i(tk)] and [tk+1,i(tk+1)].…”

Interpolation lattices in several variables

“…. , ω n , ω) are nonzero we have that α n+1 = 0 if and only if (12) holds. Lemma 6 is the key to finding functions f 0 , .…”

“…and admit an integral representation that has been given in [12]. To state this formula, we need some more notation.…”

“…By means of the simplex spline integral [X] f = we then obtain the following error formula that is valid for any f ∈ C m+1 (R n ), see Theorem 3.4 and Theorem 3.6 of [12]:…”