Lagrange polynomials over Clifford numbers

TL;DR

This paper develops Lagrange interpolation polynomials for quaternionic and Clifford algebra elements, providing new solutions for these algebraic structures using slice regular functions, with the quaternionic case being entirely novel.

Contribution

It introduces the first Lagrange interpolation polynomials for quaternions and Clifford algebras R_{0,3}, expanding the theory of polynomial interpolation in non-commutative algebras.

Findings

Complete solution for quaternionic interpolation problem

First-time interpolation approach for R_{0,3} Clifford algebra

Identification of difficulties in extending to other Clifford algebras

Abstract

We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions $H\simeq R_{0,2}$, or to the real Clifford algebra $R_{0,3}$. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of $R_{0,3}$, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases $R_{0,0}\simeq R$, $R_{0,1}\simeq C$ and the trivial case $R_{1,0}\simeq R\oplus R$, the interpolation problem on Clifford algebras $R_{p,q}$ with $(p,q)\neq(0,2),(0,3)$ seems to have some intrinsic difficulties.