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The Schur algorithm for generalized Schur functions III: J-unitary matrix polynomials on the circle

2003; Elsevier BV; Volume: 369; Linguagem: Inglês

10.1016/s0024-3795(02)00734-6

1873-1856

Daniel Alpay, T. Ya. Azizov, Aad Dijksma, Heinz Langer,

Mathematical functions and polynomials

The main result is that for J= 1 0 0 −1 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 2 k . 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.