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 21 in Operator Theory, 2026, pp 499-549 from Springer

Abstract: Abstract 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 $$2 \times 2$$ matrix polynomials, which are 1 0 0 −1 $$\begin {bmatrix} 1 & 0 \\ 0 & -1 \end {bmatrix}$$ -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 ) $$s(z)$$ on the open unit disk for which the kernel 1 − s ( z ) s ( w ) ∗ 1 − z w ∗ $$\frac {1-s(z)s(w)^*}{1-zw^*}$$ 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 ) $$n(z)$$ on the open upper half plane for which the kernel n ( z ) − n ( w ) ∗ z − w ∗ $$\frac {n(z)-n(w)^*}{z-w^*}$$ has finitely many negative squares.

Keywords: Schur transformation; Kernel with negative squares; Generalized Schur function; Minimal colligation; Schur parameter; Interpolation; J-unitary matrix polynomial; Minimal factorization; Generalized Nevanlinna function (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_41

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DOI: 10.1007/978-3-032-16356-1_41

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