Schur Analysis in an Indefinite Setting

Aad Dijksma ()
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Aad Dijksma: University of Groningen, Johann Bernoulli Institute of Mathematics and Computer Science

Chapter 13 in Operator Theory, 2015, pp 261-310 from Springer

Abstract: Abstract Schur analysis comprises topics like: the Schur transformation on the class of Schur functions (by definition, the functions which 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 1 0 0 − 1 $$\left [\begin{array}{*{10}c} 1& 0\\ 0 &-1 \end{array} \right ]$$ -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 1 − s ( z ) s ( w ) ∗ 1 − z w ∗ $$\frac{1-s(z)s(w)^{{\ast}}} {1-zw^{{\ast}}}$$ 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 n ( z ) − n ( w ) ∗ z − w ∗ $$\frac{n(z)-n(w)^{{\ast}}} {z-w^{{\ast}}}$$ has finitely many negative squares.

Keywords: Schur Analysis; Schur Transformation; Generalized Schur Functions; Schur Parameters; Schur Algorithm (search for similar items in EconPapers)
Date: 2015
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-0667-1_41

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DOI: 10.1007/978-3-0348-0667-1_41

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