ASYMPTOTICS FOR ORTHOGONAL RATIONAL FUNCTIONS

Let {alpha(n)} be a sequence of (not necessarily distinct) points in t he open unit disk, and let [GRAPHICS] (alpha(n)/\alpha(n)\ = -1 when a lpha(n) = 0. Let mu be a finite (positive) Borel measure on the unit c ircle, and let {phi(n)(z)} be orthonormal functions obtained by orthog onalization of {B(n): n = 0, 1, 2, ...} with respect to mu. Boundednes s and convergence properties of the reciprocal orthogonal functions ph in(z) = B(n)(z)phi(n)(1/zBAR) and the reproducing kernels k(n)(z, w) = SIGMA(m=0)n phi(m)(z)phi(m)(w) are discussed in the situation \alpha (n)\ less-than-or-equal-to R < 1 for all n, in particular their relati onship to the Szego condition integral(-pi)pi ln mu'(theta) dtheta > - infinity and noncompleteness in L2(mu) of the system {phi(n)(z): n = 0 , 1, 2, ...}. Limit functions of phin (z) and k(n)(z, w) are obtained . In particular , if a subsequence {alpha(n)(s)} converge to alpha, th en the subsequence {phin(s)(z)} converges to e(ilambda) square-root 1 - \alpha\2/1 - alphaBARz 1/sigma(mu)(z), lambda is-an-element-of R, w here sigma(mu)(z) = square-root 2pi exp [1/4pi integral(-pi)pi e(ithet a) + z/e(itheta) - z ln mu'(theta)dtheta]. The kernels {k(n)(z, w)} co nverge to 1/(1 - zwBAR)sigma(mu)(z)sigma(mu)(w). The results generaliz e corresponding results from the classical Szego theory (concerned wit h the polynomial situation alpha(n) = 0 for all n).