  We introduce the following linear combination interpolation problem (LCI):
Given $N$ distinct numbers $w_1,\ldots w_N$ and $N+1$ complex numbers
$a_1,\ldots, a_N$ and $c$, find all functions $f(z)$ analytic in a simply
connected set (depending on $f$) containing the points $w_1,\ldots,w_N$ such
that \[ \sum_{u=1}^Na_uf(w_u)=c. \] To this end we prove a representation
theorem for such functions $f$ in terms of an associated polynomial $p(z)$. We
first introduce the following two operations, $(i)$ substitution of $p$, and
$(ii)$ multiplication by monomials $z^j, 0\le j < N$. Then let $M$ be the
module generated by these two operations, acting on functions analytic near
$0$. We prove that every function $f$, analytic in a neighborhood of the roots
of $p$, is in $M$. In fact, this representation of $f$ is unique. To solve the
above interpolation problem, we employ an adapted systems theoretic
realization, as well as an associated representation of the Cuntz relations
(from multi-variable operator theory.) We study these operations in reproducing
kernel Hilbert space): We give necessary and sufficient condition for existence
of realizations of these representation of the Cuntz relations by operators in
certain reproducing kernel Hilbert spaces, and offer infinite product
factorizations of the corresponding kernels.
