On Multivariate Interpolation

  title={On Multivariate Interpolation},
  author={Peter J. Olver},
  journal={Studies in Applied Mathematics},
  • P. Olver
  • Published 1 February 2006
  • Mathematics
  • Studies in Applied Mathematics
A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of noncommutative quasi‐determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. 

On the Newton bivariate polynomial interpolation with applications

This work provides recursive algorithms for the computation of the Newton interpolation polynomial of a given two-variable function with known upper bounds on the degree of each indeterminate.

Extension Of Lagrange Interpolation

The aim of this paper is to construct a polynomials in space with error tends to zero by using Gramer's formula.

Case study in bivariate Hermite interpolation

Multivariate Polynomial Interpolation in Newton Forms

Techniques of univariate Newton interpolating polynomials are extended to multivariate data points by different generalizations and practical algorithms. The Newton basis format, with divided-diffe...

A Simple Expression for Multivariate Lagrange Interpolation

We derive a simple formula for constructing the degree n multinomial function which interpolates a set of ( n+m n ) points in Rm+1, when the function is unique. The formula coincides with the

Multivariate polynomial interpolation on lower sets

On the irreducibility of generalized Vandermonde determinants

We find geometric and arithmetic conditions on a finite set of integer exponents and the characteristic of a field k in order to characterize the irreducibility of the determinant of the generic

Christoffel transformations for multivariate orthogonal polynomials

Learning algebraic varieties from samples

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic

Geometric Integration via Multi-space

The method of invariantization is based on the equivariant moving frame theory applied to prolonged symmetry group actions on multi-space, which has been proposed as the proper geometric setting for numerical analysis.



On the Sauer-Xu formula for the error in multivariate polynomial interpolation

Use of a new notion of multivariate divided difference leads to a short proof of a formula by Sauer and Xu for the error in interpolation, by polynomials of total degree? n in d variables, at a

Multivariate Birkhoff Interpolation

Univariate interpolation.- Basic properties of Birkhoff interpolation.- Singular interpolation schemes.- Shifts and coalescences.- Decomposition theorems.- Reduction.- Examples.- Uniform Hermite

A Constructive Approach to Kergin Interpolation in R(k).

Abstract : Very little seems to be known about polynomial interpolation of multivariate functions. However, Kergin recently established the existence and uniqueness of a natural extension of

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,

Multivariate Differences, Polynomials, and Splines

We generalize the univariate divided difference to a multivariate setting by considering linear combinations of point evaluations that annihilate the null space of certain differential operators. The

Quasideterminants, I

0. Introduction 1. A general theory and main identities 2. Important example: quaternionic quasideterminants 3. Noncommutative determinants 4. Noncommutative Plücker and flag coordinates 5. Bezout

Relations Between Roots and Coefficients, Interpolation and Application to System Solving

An algebraic framework to represent zero-dimensional algebraic systems is proposed and new interpolation formulae are given to develop a generalization of Weierstrass?s method to the multivariate systems.

A Variational Complex for Difference Equations

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved and a variational complex for difference equations is constructed and is proved to be locally exact.