Rational Matrix Functions with J-Unitary Values on the Imaginary Line

Abstract

In this chapter we consider rational matrix functions U(z) which have J-unitary values on the imaginary axis, i.e. (U(z))*JU(z) = J for any regular point z of U in i ℝ. Here J is a signature matrix, namely J = J* and J 2 = I. For this class of matrix functions we analyze the null-pole structure, the special properties of realizations for this class, and the appropriate interpolation problems. We also consider here the interpolation problem with standard local data and its derivatives. These problems are also matricially homogeneous, with the understanding that one is permitted to multiply by J-unitary matrices. The coupling matrix which appeared in the general interpolation problems from Chapter 4 now is Hermitian and plays an extra role. The case J = I hasspecial features and is discussed separately.