# On Multivariate Interpolation

@article{Olver2006OnMI, title={On Multivariate Interpolation}, author={Peter J. Olver}, journal={Studies in Applied Mathematics}, year={2006}, volume={116} }

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.

## 120 Citations

### On the Newton bivariate polynomial interpolation with applications

- Mathematics, Computer ScienceMultidimens. Syst. Signal Process.
- 2014

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

- Mathematics
- 2015

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

### Multivariate Polynomial Interpolation in Newton Forms

- MathematicsSIAM Rev.
- 2019

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

- Mathematics
- 2008

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…

### On the irreducibility of generalized Vandermonde determinants

- Mathematics
- 2008

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

- MathematicsJ. Approx. Theory
- 2018

### Learning algebraic varieties from samples

- MathematicsRevista Matemática Complutense
- 2018

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…

### Bivariate Hermite-Birkhoff interpolation and Vandermonde determinants

- MathematicsNumerical Algorithms
- 2005

The concepts of Vandermonde determinant and confluent Vandermonde determinant are extended to the multidimensional setting by relating them to multivariate interpolation problems. With an approach…

## References

SHOWING 1-10 OF 86 REFERENCES

### Multivariate divided differences and multivariate interpolation of Lagrange and Hermite type

- Mathematics
- 1982

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

- MathematicsMath. Comput.
- 1996

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

- Mathematics
- 1992

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).

- Mathematics
- 1978

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

- Mathematics
- 1995

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,…

### Factorization of differential operators, quasideterminants, and nonabelian Toda field equations

- Mathematics
- 1997

We integrate nonabelian Toda field equations for root systems of types A, B, C, for functions with values in any associative algebra. The solution is expressed via quasideterminants. In the appendix…

### Multivariate Differences, Polynomials, and Splines

- Mathematics
- 1996

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

- Mathematics
- 1997

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…