# An application of biregularity to quaternionic Lagrange interpolation

Perotti, Alessandro (2008) An application of biregularity to quaternionic Lagrange interpolation. Conference Proceedings ; 1048 . American Institute of Physics, pp. 691-694.

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## Abstract

We revisit the concept of totally analytic variable of one quaternionic variable introduced by Delanghe \cite Delanghe} and its application to Lagrange interpolation by G\"uerlebeck and Spr\"ossig \cite{GS}. We consider left-regular functions in the kernel of the Cauchy-Riemann operator $\mathcal D=2\left(\frac{\partial}{\partial{\bar z_1}}+j\frac{\partial}{\partial{\bar z_2}}\right)=\frac{\partial}{\partial{x_0}}+i\frac{\partial}{\partial{x_1}}+j\frac{\partial}{\partial{x_2}}-k\frac{\partial}{\partial{x_3}}.$ For every imaginary unit $p\in {\Sp}^2$, let ${\CC}_p=\langle 1,p\rangle\simeq {\CC}$ and let $J_p=p_1J_1+p_2J_2+p_3J_3$ be the corresponding complex structure on ${\HH}$. We identify totally regular variables with real--affine holomorphic functions from $({\HH},J_p)$ to $({\CC}_p,L_p)$, where $L_p$ is the complex structure defined by left multiplication by $p$. We then show that every $J_p$--biholomorphic map, which is always a biregular function, gives rise to a Lagrange interpolation formula at any set of distinct points in ${\HH}$. Publisher version at: http://link.aip.org/link/?APCPCS/1048/691/1

Item Type: Conference paper Mathematics Q Science > QA Mathematics > QA152 Algebra Quaternionic regular function, biregular function, Lagrange interpolation 20 Aug 2009

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