A new parametrization of rational inner functions of fixed degree: Schur parameters and realizations
A new parametrization of rational inner \(p\times p\) matrix-valued functions \(Q\) of a given McMillan degree \(n\) which satisfy the condition \(Q(\infty)=I_p\) is presented. The manifold \({\mathcal I}^p_n(\infty)\) of such functions can be characterized by constructing an appropriate atlas by means of the tangential Schur algorithm which gives, for \(Q\in {\mathcal I}^p_n(\infty)\), a sequence of inner matrix-valued functions \(Q^{(n)}=Q, Q^{(n-1)}, \ldots ,Q^{(1)},Q^{(0)}=I_p\) of decreasing McMillan degree that satisfy the interpolation conditions \(u_i^*Q^{(i)}(\omega_i)=v_i^*\), where \(\omega_i\in\Pi^+\) (the right half-plane), \(u_i\in {\mathbb C}^p\) and \(v_i\in {\mathbb C}^p\) are column vectors such that \(\|v_i\|<\|u_i\|\;(i=1,\ldots ,n)\). The coordinates of \(Q\) in the chart are vectors \(v_i\;(i=1,\ldots ,n)\) provided \(\omega =(\omega_1,\ldots ,\omega_n)\) and \(u=(u_1,\ldots ,u_n)\) are given. The Schur algorithm yields a minimal state-space realization of \(Q\) which has a diagonal observability Grammian. However, the latter will change together with \(v_i\), which is unsatisfactory from the computational point of view. To avoid this difficulty, the authors propose a new way of parametrizing, in which the observability Grammian is assumed to be a constant for each chart, and then there exists, under proper assumptions, unique points \(\omega_i\) in the interpolation problems above. The advantages of such a parametrization are demonstrated in the application to the \(L^2\) rational approximation problem. Two examples of this parametrization are given in the case of inner real rational \(2\times 2\) matrices of McMillan degree 1, and in the case of scalar inner real rational functions of McMillan degree 2.
scientific article; zbMATH DE number 1495920