Rational Hermite–Fejér interpolation

Abstract. In this paper we construct rational orthogonal systems with respect to the normalized area measure on the unit disc. The generating system is a collection of so called elementary rational functions. In the one dimensional case an explicit formula exists for the corresponding Malmquist–Takenaka functions involving the Blaschke functions. Unfortunately, this formula has no generalization for the case of the unit disc, which justifies our investigations. We focus our attention for such special pole combinations, when an explicit numerical process can be given. In [4] we showed, among others, that if the poles of the elementary rational functions are of order one, then the orthogonalization is naturally related with an interpolation problem. Here we take systems of poles which are of order both one and two. We show that this case leads to an Hermite–Fejér type interpolation problem in a subspace of rational functions. The orthogonal projection onto this subspace is calculated and also the basic Hermite–Fejér interpolation functions are provided. We show that by means of the given process effective algorithms can be constructed for approximations of surfaces.