$q$-Rational functions and interpolation with complete Nevanlinna Pick Kernels

In this paper we introduce the concept of matrix-valued $q$-rational functions. In comparison to the classic case we give different characterizations with principal emphasise on realizations and discuss algebraic manipulations. We also study the concept of Schur multipliers and complete Nevanlinna Pick kernels in this context and provide first applications in terms of an interpolation problem using Schur multipliers and complete Nevanlinna Pick kernels.