A new realization of rational functions, with applications to linear combination interpolation, the Cuntz relations and kernel decompositions

We introduce the following linear combination interpolation problem (LCI), which in case of simple nodes reads as follows: given N distinct numbers w(1),... w(N) and N+1 complex numbers a(1),..., a(N) and c, find all functions f (z) analytic in an open set (depending on f) containing the points w(1),..., w(N) such that Sigma(N)(u=1) a(u) f(w(u)) = c. To this end, we prove a representation theorem for such functions f in terms of an associated polynomial p(z). We give applications of this representation theorem to realization of rational functions and representations of positive definite kernels.