Schur Analysis in an Indefinite Setting

Schur analysis comprises topics like: the Schur transformation on the class of Schur functions (by definition, the functions that are holomorphic and bounded by 1 on the open unit disk) and the Schur algorithm, Schur parameters and approximation, interpolation problems for Schur functions, factorization of rational 2 × 2 matrix polynomials, which are \(\begin {bmatrix} 1 & 0 \\ 0 & -1 \end {bmatrix}\) -unitary on the unit circle, and a related inverse scattering problem. This note contains a survey of indefinite versions of these topics related to the class of scalar generalized Schur functions. These are the meromorphic functions s(z) on the open unit disk for which the kernel \(\frac {1-s(z)s(w)^*}{1-zw^*}\) has finitely many negative squares. We also review a generalization of the Schur transformation to classes of functions on a general domain one of which is the class of scalar generalized Nevanlinna functions. These are the meromorphic functions n(z) on the open upper half plane for which the kernel \(\frac {n(z)-n(w)^*}{z-w^*}\) has finitely many negative squares.