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

Abstract

Let {[Phi]n}n[greater-or-equal, slanted]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 of monic OPUC whose sequence of Verblunsky coefficients iswhere b1,b2,...,bN-1 are N-1 fixed real numbers such that bj[set membership, variant](-1,1) for all j=1,2,...,N-1, so that is also orthogonal with respect to a symmetric and finite positive Borel measure on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to {[Phi]n}n[greater-or-equal, slanted]0 and (by Szegö's transformation) are related by some polynomial mapping, giving rise to a one-to-one correspondence between the monic OPUC on the unit circle and a pair of monic OPRL on (a subset of) the interval [-1,1]. In particular we prove thatd[mu][small tilde]([theta])=[zeta]N-1([theta])sin[theta]sin[theta]N([theta])d[mu]([theta]N([theta]))[theta]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,b1,...,bN-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 Szegö's transformation may be applicable.