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

Let {Φn}n⩾0{Φn}n⩾0 be a sequence of monic orthogonal polynomials on the unit circle (OPUC) with respect to a symmetric and finite positive Borel measure dμdμ on [0,2π][0,2π] and let -1,α0,α1,α2,…-1,α0,α1,α2,… be the associated sequence of Verblunsky coefficients. In this paper we study the sequence {Φ˜n}n⩾0 of monic OPUC whose sequence of Verblunsky coefficients is-1,-b1,-b2,…,-bN-1,α0,bN-1,…,b2,b1,α1,-b1,-b2,…,-bN-1,α2,bN-1,…,b2,b1,α3,…where b1,b2,…,bN-1b1,b2,…,bN-1 are N-1N-1 fixed real numbers such that bj∈(-1,1)bj∈(-1,1) for all j=1,2,…,N-1j=1,2,…,N-1, so that {Φ˜n}n⩾0 is also orthogonal with respect to a symmetric and finite positive Borel measure dμ˜ on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {Φn}n⩾0{Φn}n⩾0 and {Φ˜n}n⩾0 (by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC {Φ˜n}n⩾0 on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1,1][-1,1]. In particular we prove thatdμ˜(θ)=ζN-1(θ)sinθsinϑN(θ)dμ(ϑN(θ))ϑN′(θ),supported on (a subset of) the union of 2N2N intervals contained in [0,2π][0,2π] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, ζN-1ζN-1 and cosϑNcosϑN are algebraic polynomials in cosθcosθ of degrees N-1N-1 and N   (respectively) defined only in terms of α0,b1,…,bN-1α0,b1,…,bN-1. This measure induces a measure on the unit circle supported on the union of 2N2N arcs, pairwise symmetric with respect to the real axis. The restriction to symmetric measures (or real Verblunsky coefficients) is needed in order that Szegö's transformation may be applicable.