Polynomials and Vandermonde Matrices over the Field of Quaternions

It is known that the space of real valued, continuous functions C(B) over a multidimensional compact domain B ⊂ R , k ≥ 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 ⊂ C. Nevertheless, in this situation Haar spaces V ⊂ C(B) exist. We are concerned here with the case of quaternionic valued, continuous functions C(B) where B ⊂ 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.