The Schur Algorithm for Generalized Schur Functions III: Reproducing Kernels
The main result is that for
every J-unitary 2×2-matrix polynomial on the unit circle is an essentially unique product of elementary J-unitary 2×2-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, Birkhäuser Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1–54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkhäuser Verlag, Basel, 2000, p. 1; Monatshefte für Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions.
D. Alpay, T. Azizov, A. Dijksma and H. Langer. The Schur algorithm for generalized Schur functions III: Reproducing kernels. Linear Algebra and its applications, vol. 369 (2003) 113-144.