1. The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle
- Author
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Tomas Ya. Azizov, Aad Dijksma, Heinz Langer, Daniel Alpay, and Systems, Control and Applied Analysis
- Subjects
Numerical Analysis ,Algebra and Number Theory ,Schur's lemma ,generalized Schur functions ,Schur algebra ,Schur polynomial ,Schur's theorem ,Jack function ,Combinatorics ,minimal factorizations ,Schur decomposition ,reproducing kernel Pontryagin spaces ,generalized Schur algorithm ,Schur complement ,Discrete Mathematics and Combinatorics ,kernels with negative squares ,Geometry and Topology ,elementary J-unitary matrix polynomials ,Schur product theorem ,Mathematics - Abstract
The main result is that forJ = ((1)(0) (0)(-1))every J-unitary 2 x 2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2 x 2-matrix polynomials which are either of degree 1 or 2k. This is shown by means of the generalized Schur transformation introduced in [Ann. Inst. Fourier 8 (1958) 211; Ann. Acad. Sci. Fenn. Ser. A I 250 (9) (1958) 1-7] and studied in [Pisot and Salem Numbers, Birkhauser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1-54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhauser Verlag, Basel, 2000, p. 1; Monatshefte fur Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions. (C) 2003 Elsevier Science Inc. All fights reserved.
- Published
- 2003
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