THE ASYMPTOTICS OF A CONTINUOUS ANALOG OF ORTHOGONAL POLYNOMIALS

Szego polynomials are associated with weight functions on the unit cir cle. M. G. Krein introduced a continuous analogue of these, a family o f entire functions of exponential type associated with a weight functi on on the real line. An investigation of the asymptotics of the resolv ent kernel of sin(x - y)/pi(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 in volving the above-mentioned kernel, questions arising in the theory of random matrices. (C) 1994 Academic Press, Inc.