ORTHOGONAL RATIONAL FUNCTIONS WITH POLES ON THE UNIT-CIRCLE

Let {alpha(n)} be a sequence of (not necessarily distinct) points on t he unit circle T= {z is-an-element-of C: Absolute value of z = 1). Set L(n) =Span {1, 1/omega1, ..., 1/omega(n)}, L = or n=0(infinity) L(n), where we have used the notation omega(n) = PI(k=1)n (z - alpha(k)). L et M be a positive linear functional defined on the space L . L with M (R) real for functions that are real on T. Define [R, S] = M(R(z) S(1/ 2)) for R, SBAR is-an-element-of L. (In particular if M is given as M( R) integral-pi(pi) R(e(itheta)) dmu(theta) for some measure mu, then [ R, S] = integral-pi(pi)R(e(itheta)) S(e(itheta)dmu(theta)BAR.) Let the orthogonal system {phi(n)} be obtained from {1/omega(n)} by orthogona lization. Three-term recurrence relations, quadrature formulas, moment theory, and interpolation properties connected with the functional M and the system {phi(n)} are discussed. (C) 1994 Academic Press, Inc.