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 interval s 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 exist s a real trigonometric polynomial tau(N)(phi) with maximal number, i.e ., N + l, of extremal points on E(l). The associated algebraic polynom ial F-N(z)=z(N/2)tau(N)(z), z=e(i phi), is called the complex Chebyshe v polynomial. Then it is shown that polynomials orthogonal on E(l) hav e 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 comp lex Chebyshev polynomial on E(l). Let us point out in this connection that Geronimus has shown that orthogonal polyno mials generated by per iodic reflection coefficients of absolute value less than 1 are orthog onal with respect to a measure of the above type. Furthermore, we deri ve explicit representations of the corresponding orthogonal polynomial s with the help of the complex Chebyshev polynomials. Finally, we prov ide a characterization of those definite functionals to which orthogon al polynomials with periodic reflection coefficients of modulus unequa l to one are orthogonal. (C) 1996 Academic Press, Inc.