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

The main result is that forJ=<fen><cp type="lpar" STYLE="S"><ar><r><c ca="c" CSPAN="1" RSPAN="1" RA="T">1</c><c ca="c" CSPAN="1" RSPAN="1" RA="T">0</c></r><r><c ca="c" CSPAN="1" RSPAN="1" RA="T">0</c><c ca="c" CSPAN="1" RSPAN="1" RA="T">−1</c></r></ar><cp type="rpar" STYLE="S"></fen>every <f>J</f>-unitary <f>2×2</f>-matrix polynomial on the unit circle is an essentially unique product of elementary <f>J</f>-unitary <f>2×2</f>-matrix polynomials which are either of degree <f>1</f> or <f>2k</f>. 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, Birkha¨user Verlag, Basel, 1992; Philips J. Res. 41 (1) (1986) 1–54], and also in the first two parts [Operator Theory: Adv. Appl. 129, Birkha¨user Verlag, Basel, 2000, p. 1; Monatshefte fu¨r Mathematik, in press] of this series. The essential tool in this paper are the reproducing kernel Pontryagin spaces associated with generalized Schur functions. [Copyright &y& Elsevier]