Interpolation Problems for Certain Classes of Slice Hyperholomorphic Functions

A general interpolation problem (which includes as particular cases the Nevanlinna–Pick and Carathéodory–Fejér interpolation problems) is considered in two classes of slice hyperholomorphic functions of the unit ball of the quaternions. In the Hardy space of the unit ball we present a Beurling–Lax type parametrization of all solutions, and the formula for the minimal norm solution. In the class of functions slice hyperholomorphic in the unit ball and bounded by one in modulus there (that is, in the class of Schur functions in the present framework) we present a necessary and sufficient condition for the problem to have a solution, and describe the set of all solutions in the indeterminate case.