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

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.