Lagrange interpolation problem for quaternion polynomials

Vladimir Bolotnikov

doi:10.1016/j.crma.2014.05.008

Mathematical analysis/Complex analysis

Lagrange interpolation problem for quaternion polynomials

Presented by: the Editorial Board

Vladimir Bolotnikov ¹

¹ Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 577-581.

Abstract
Résumé

Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated “from the left” and “from the right”. If the interpolation problem involves interpolation conditions of the same (left or right) type, the results are very much similar to the complex case: a consistent problem has a unique solution of a low degree (less than the number of interpolation conditions imposed), and the solution set of the homogeneous problem is an ideal in the ring $H [z]$ . The problem containing both “left” and “right” interpolation conditions is quite different: there may exist infinitely many low-degree solutions and the solution set of the homogeneous problem is a quasi-ideal in $H [z]$ .

La théorie de l'interpolation pour les polynômes complexes est bien compris. Dans le cadre des quaternions non commutatifs, les polynômes peuvent être évalués « de la gauche » et « de la droite ». Si le problème d'interpolation implique des conditions d'interpolation du même type (gauche ou droite), les résultats sont très similaires au cas complexe : un problème constant a une solution unique d'un faible degré (moins que le nombre de conditions d'interpolation qui sont imposées), et l'ensemble du problème homogène est un idéal de l'anneau $H [z]$ . Le problème qui contient des conditions d'interpolation à la fois « gauches » et « droites » est tout à fait différent : il peut exister une infinité de solutions d'un faible degré, et l'ensemble des solutions du problème homogène est un quasi-idéal de $H [z]$ .

Received: 2014-02-14
Accepted: 2014-05-14
Published online: 2014-06-19

DOI: 10.1016/j.crma.2014.05.008

Author's affiliations:

Vladimir Bolotnikov ¹

¹ Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA

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     title = {Lagrange interpolation problem for quaternion polynomials},
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Vladimir Bolotnikov. Lagrange interpolation problem for quaternion polynomials. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 577-581. doi : 10.1016/j.crma.2014.05.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.05.008/

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