Generalizations of the Nevanlinna-Pick interpolation Problem

This paper aims at generalizing the well-known Nevanlinna-Pick interpolation problem by considering additional constraints. The first type of constraints we consider requires the interpolation function to be of a given degree. Several results are provided for different degree constraints. These results offer feasibility tests via linear matrix inequalities. We have identified a number of degree constraints for which the feasibility tests are exact. For other degree constraints, we offer a relaxation scheme for checking the feasibility. The second type of constraints we study is about spectral zero assignment, which demands the zeros of the spectral factorization of the interpolation function to be at given locations. This problem can be solved using an iterative algorithm by Byrnes, Georgiou and Linquist. However, we provide a much faster iterative algorithm for this problem, although a proof of convergence is yet to be offered.