Geodesic
Polynomials and Vandermonde matrices over the field of quaternions
Electronic transactions on numerical analysis, Tome 36 (2010)
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},
     year = {2010},
     volume = {36},
     zbl = {1196.11154},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2010__36__a11/}
}
TY  - JOUR
AU  - Opfer,  Gerhard
TI  - Polynomials and Vandermonde matrices over the field of quaternions
JO  - Electronic transactions on numerical analysis
PY  - 2010
VL  - 36
UR  - http://geodesic.mathdoc.fr/item/ETNA_2010__36__a11/
LA  - en
ID  - ETNA_2010__36__a11
ER  - 
%0 Journal Article
%A Opfer,  Gerhard
%T Polynomials and Vandermonde matrices over the field of quaternions
%J Electronic transactions on numerical analysis
%D 2010
%V 36
%U http://geodesic.mathdoc.fr/item/ETNA_2010__36__a11/
%G en
%F ETNA_2010__36__a11
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/