A basic interpolation problem for generalized Schur functions and coisometric realizations Academic Article uri icon

abstract

  • The basic interpolation problem for Schur functions is: Find all Schur functions s(z)for which s (0) has a given value. In this paper we consider the same basic interpolation problem but now for the class of generalized Schur functions with finitely many negative squares which are holomorphic at z = 0. In Section3 its solutions are given by three fractional linear transformations in which the main parameter runs through a subset of the class of generalized Schur functions. A generalized Schur function can be written as the characteristic function of a minimal coisometric colligation with a Pontryagin state space. In the second part of this paper we describe the colligation of a solution s(z) of the basic interpolation problem for generalized Schur functions in terms of the colligation of the corresponding parameter function and the interpolation data. First we consider the canonical coisometric realization of s(z) in which the state space is the reproducing kernel Pontryagin space with kernel \(\frac{{1 - s\left( z \right)s{{\left( \omega \right)}^*}}}{{1 - z{\omega ^*}}}\);see for example [6]. In the final section we follow a direct approach more in line with [2, 3].

publication date

  • January 1, 2003