On the “redundant” null-pairs of functions connected by a general Linear Fractional Transformation

Abstract

We investigate here the interpolation conditions connected to an interpolating function \(Q\) obtained as a Linear Fractional Transformation of another function \(S\). In general, the degree of \(Q\) is equal to the number of interpolating conditions plus the degree of \(S\). We show that, if the degree of \(Q\) is strictly less that this quantity, there is a number of complementary interpolating conditions which has to be satisfied by \(S\). This induces a partitioning of the interpolating conditions in two sets. We consider here the case where these two sets are not necessarily disjoint. The reasoning can also be reversed (i.e. from \(S\) to \(Q\)). To derive the above results, a generalized interpolation problem, which relaxes the usual assumptions on disjointness of the interpolation nodes and the poles of the interpolant, is formulated and solved.