Orthogonal Polynomials on Arcs of the Unit Circle: II. Orthogonal Polynomials with Periodic Reflection Coefficients

Abstract

First we give necessary and sufficient conditions on a set of intervalsEl=∪lj=1[ϕ2j−1, 2j], 1<…<2l and 2l−ϕ1⩽2π, such that onElthere exists a real trigonometric polynomialτN(ϕ) with maximal number, i.e.,N+l, of extremal points onEl. The associated algebraic polynomialTN(z)=zN/2τN(z),z=eiϕ, is called the complex Chebyshev polynomial. Then it is shown that polynomials orthogonal onElhave periodic reflection coefficients if and only if they are orthogonal onElwith respect to a measure of the form[formula]certain point measures, whereAis a real trigonometric polynomial with no zeros onEland there exists a complex Chebyshev polynomial onEl. Let us point out in this connection that Geronimus has shown that orthogonal polynomials generated by periodic reflection coefficients of absolute value less than 1 are orthogonal with respect to a measure of the above type. Furthermore, we derive explicit representations of the corresponding orthogonal polynomials with the help of the complex Chebyshev polynomials. Finally, we provide a characterization of those definite functionals to which orthogonal polynomials with periodic reflection coefficients of modulus unequal to one are orthogonal.