Orthogonal polynomials on arcs of the unit circle .2. Orthogonal polynomials with periodic reflection coefficients

First we give necessary and sufficient conditions on a set of intervals E(1)= U-j=1(l) [phi(2j-1), phi(2j)], phi(1) < ... < phi(2l) and phi(2l) - phi(1) less than or equal to 2 pi, such that on E(I) there exists a real trigonometric polynomial tau(N)(phi) with maximal number, i.e., N + l, of extremal points on E(l). The associated algebraic polynomial F-N(z)=z(N/2)tau(N)(z), z=e(i phi), is called the complex Chebyshev polynomial. Then it is shown that polynomials orthogonal on E(l) have periodic reflection coefficients if and only if they are orthogonal on E(l) with respect to a measure of the form root-Pi(j=1)(2l)sin((phi - phi(j))/2)/ A(phi)d phi+ certain point measures, where A is a real trigonometric polynomial with no zeros on E(l) and there exists a complex Chebyshev polynomial on E(l). Let us point out in this connection that Geronimus has shown that orthogonal polyno mials 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. (C) 1996 Academic Press, Inc.