Lagrange polynomials over Clifford numbers

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.