Volume 36, pp. 9-16, 2009-2010.

Polynomials and Vandermonde matrices over the field of quaternions

Gerhard Opfer

Abstract

It is known that the space of real valued, continuous functions $C(B)$ over a multidimensional compact domain $B\subset {\mathbf{R}}^k,k\ge 2$ does not admit Haar spaces, which means that interpolation problems in finite dimensional subspaces $V$ of $C(B)$ may not have a solutions in $C(B)$. The corresponding standard short and elegant proof does not apply to complex valued functions over $B\subset \mathbf{C}$. Nevertheless, in this situation Haar spaces $V\subset C(B)$ exist. We are concerned here with the case of quaternionic valued, continuous functions $C(B)$ where $B\subset \mathbf{H}$ and $\mathbf{V}$ denotes the skew field of quaternions. Again, the proof is not applicable. However, we show that the interpolation problem is not unisolvent, by constructing quaternionic entries for a Vandermonde matrix $\mathbf{V}$ such that $\mathbf{V}$ will be singular for all orders $n>2$. In addition, there is a section on the exclusion and inclusion of all zeros in certain balls in $\mathbf{H}$ for general quaternionic polynomials.

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Key words

Quaternionic interpolation polynomials, Vandermonde matrix in quaternions, location of zeros of quaternionic polynomials

AMS subject classifications

11R52, 12E15, 12Y05, 65D05

Links to the cited ETNA articles

[8]	Vol. 26 (2007), pp. 82-102 Drahoslava Janovská and Gerhard Opfer: Computing quaternionic roots by Newton's method

ETNA articles which cite this article

Vol. 44 (2015), pp. 660-670 Gerhard Opfer: Polynomial interpolation in nondivision algebras

Vol. 54 (2021), pp. 128-149 Sk. Safique Ahmad, Istkhar Ali, and Ivan Slapničar: Perturbation analysis of matrices over a quaternion division algebra