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

Abstract

Let { Φ 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 μ on [ 0 , 2 π ] and let - 1 , α 0 , α 1 , α 2 , … 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 ,- b 1 ,- b 2 , … ,- b N - 1 , α 0 , b N - 1 , … , b 2 , b 1 , α 1 ,- b 1 ,- b 2 , … ,- b N - 1 , α 2 , b N - 1 , … , b 2 , b 1 , α 3 , … where b 1 , b 2 , … , b N - 1 are N - 1 fixed real numbers such that b j ∈ ( - 1 , 1 ) for all j = 1 , 2 , … , N - 1 , so that { Φ ˜ n } n ⩾ 0 is also orthogonal with respect to a symmetric and finite positive Borel measure d μ ˜ on the unit circle. We show that the sequences of monic orthogonal polynomials on the real line (OPRL) corresponding to { Φ n } 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 ] . In particular we prove that d μ ˜ ( θ ) = ζ N - 1 ( θ ) sin θ sin ϑ N ( θ ) d μ ( ϑ N ( θ ) ) ϑ N ′ ( θ ) , supported on (a subset of) the union of 2 N intervals contained in [ 0 , 2 π ] such that any two of these intervals have at most one common point, and where, up to an affine change in the variable, ζ N - 1 and cos ϑ N are algebraic polynomials in cos θ of degrees N - 1 and N (respectively) defined only in terms of α 0 , b 1 , … , b N - 1 . This measure induces a measure on the unit circle supported on the union of 2 N 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.