Orthogonal polynomials on arcs of the unit circle, 1

Let E(1)=boolean OR(j=1)(1) [phi(2j-1), phi(2j)]subset of or equal to[0, 2 pi], R(phi)=Pi(j=1)(21) sin((phi-phi(j))/2) and 1/r(phi)=(-1)(j)/root\R(phi)\ for phi is an element of(phi(2j-1), phi(2j)). Furthermore let V, W be arbitrary real trigonometric polynomials such that R = V W and let A(phi) be a real trigonometric polynomial which has no zero in E(1). First we derive an explicit representation of the Caratheodory function associated with f(phi;W) = W(phi/A(phi) r(phi) on E(1). With the help of this result the polnomials (P-n(z), which are orthogonal on the set of arcs Gamma(E1): = {e(i phi):phi is an element of E(1)} with respect to f(phi; W), are completely characterized by a quadratic equation. (In fact a more general case including Dirac-mass points is considered.) This characterization is the basis of ail of our further investigations on polynomials orthogonal on several arcs as the description of that measures which generate orthogonal polynomials with periodic or asymptotically periodic reflection coefficients, the explicit representation of the orthogonality measure of the associated polynomials, the asymptotic representation of polynomials orthogonal on Gamma(E1), etc. (C) 1996 Academic Press, Inc.