Geodesic
Polynomials and Vandermonde matrices over the field of quaternions
Electronic transactions on numerical analysis, Tome 36 (2010).

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: It is known that the space of real valued, continuous functions $C(B)$ over a multidimensional compact domain B $\subset $Rk , k $\geq 2$ 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 $\subset C$. 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 $\subset H$ and H 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 V such that 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 H for general quaternionic polynomials.
Classification : 11R52, 12E15, 12Y05, 65D05
Keywords: quaternionic interpolation polynomials, Vandermonde matrix in quaternions, location of zeros of quaternionic polynomials 
@article{ETNA_2010__36__a11,
     author = {Opfer, Gerhard},
     title = {Polynomials and {Vandermonde} matrices over the field of quaternions},
     journal = {Electronic transactions on numerical analysis},
     publisher = {mathdoc},
     volume = {36},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2010__36__a11/}
}
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Opfer, Gerhard. Polynomials and Vandermonde matrices over the field of quaternions. Electronic transactions on numerical analysis, Tome 36 (2010). http://geodesic.mathdoc.fr/item/ETNA_2010__36__a11/