Orthogonal polynomials on the unit circle via a polynomial mapping on the real line

摘要

Let {Phi(n)}(n) >= 0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure d mu on [0, 2 pi] and let - 1 alpha(0), alpha(1), alpha(2).... be the associated sequence of Verblunsky coefficients. In this paper we study the sequence {(Phi) over tilde (n)}(n) >= 0 of monic OPUC whose sequence of Verblunsky coefficients is -1, -b(1), -b(2),..., -b(N-1), alpha(0), b(N-1),..., b(2), b(1), alpha(1), -b(1), -b(2), ... , -b(N-1), alpha(2), b(N-1), ... , b(2), b(1), alpha(3), ... where b(1), b(2),..., b(N-1) are N-1 fixed real numbers such that b(j) is an element of (-1, 1) for all j = 1, 2,..., N-1, so that {(Phi) over tilde (n)}(n) >= 0 is also orthogonal with respect to a symmetric and finite positive Borel measure d (mu) over tilde on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {Phi(n)}(n) >= 0 and {(Phi) over tilde (n)}(n) >= 0 (by Szego's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC {(Phi) over tilde (n)}(n) >= 0 on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1, 1]. In particular we prove that d (mu) over tilde(theta) = |zeta(N-1)(theta)| |sin theta/sin I-N(theta)| d mu(I-N (theta))/I'(N)(theta), supported on (a subset of) the union of 2N intervals contained in [0, 2 pi] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, zeta(N-1) and cos theta(N) are algebraic polynomials in cos theta of degrees N-1 and N (respectively) defined only in terms of alpha(0), b(1), ... , b(N-1). This measure induces a measure on the unit circle supported on the union of 2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szego's transformation may be applicable. (C) 2007 Published by Elsevier B.V.