Three-dimensional interpolation with monogenic polynomials

One of the open problems in hypercomplex analysis is the interpolation of monogenic functions by monogenic polynomials. We consider the case of monogenic functions defined in a domain of with values in the algebra of quaternions. The idea is to interpolate these functions by a special system of monogenic polynomials, the so-called pseudo complex polynomials. Quaternionic analysis shows a lot of analogies to complex analysis and the monogenic functions play the role of holomorphic functions. Due to the non-commutative multiplication a fundamental theorem of algebra is not available, and standard tools from linear algebra cannot be applied in the usual way. So, the main goal of this paper is to show that the interpolation problem is solvable for arbitrarily given interpolation nodes and general interpolation data.