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 (Formula presented.) distinct numbers (Formula presented.) and (Formula presented.) complex numbers (Formula presented.) and (Formula presented.) , find all functions (Formula presented.) analytic in an open set (depending on (Formula presented.) ) containing the points (Formula presented.) such that (Formula presented.) To this end, we prove a representation theorem for such functions (Formula presented.) in terms of an associated polynomial (Formula presented.). We give applications of this representation theorem to realization of rational functions and representations of positive definite kernels.