The Asymptotics of a Continuous Analogue of Orthogonal Polynomials

Szegö polynomials are associated with weight functions on the unit circle. M. G. Krein introduced a continuous analogue of these, a family of entire functions of exponential type associated with a weight function on the real line. An investigation of the asymptotics of the resolvent kernel of sin(x − y)/π(x − y) on [0, s] leads to questions of the asymptotics of the Krein functions associated with the characteristic function of the complement of the interval [−1, 1]. Such asymptotics are determined here, and this leads to answers to certain questions involving the above-mentioned kernel, questions arising in the theory of random matrices.