Orthogonal Polynomials on Arcs of the Unit Circle, I Original Research Article

LetEl=∪lj=1 [ϕ2j−1, ϕ2j]⊆[0, 2π], R(ϕ)=∏2lj=1 sin((ϕ−ϕj)/2) and[formula]forϕ∈(ϕ2j−1, ϕ2j). Furthermore let V, W be arbitrary real trigonometric polynomials such that R=VW and let A(ϕ) be a real trigonometric polynomial which has no zero inEl. First we derive an explicit representation of the Caratheodory function associated withf(ϕ; W)=W(ϕ)/A(ϕ) r(ϕ) onEl. With the help of this result the polynomialsPn(z), which are orthogonal on the set of arcsΓEl :={eiϕ:emsp14;ϕ∈El} with respect tof(ϕ; 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 all 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 onΓEl, etc.