PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 131, Number 6, Pages 1771–1776 S 0002-9939(02)06706-0 Article electronically published on October 1, 2002 MASS POINTS OF MEASURES ON THE UNIT CIRCLE AND REFLECTION COEFFICIENTS LEONID GOLINSKII (Communicated by Andreas Seeger) Abstract. Measures on the unit circle and orthogonal polynomials are com- pletely determined by their reflection coefficients through the Szeg˝o recur- rences. We find the conditions on the reflection coefficients which provide the lack of a mass point at ζ = 1. We show that the result is sharp in a sense. 1. Introduction Let P be the set of all probability measures µ on the unit circle T = {|ζ | =1} with infinite support. The latter is defined as the smallest closed set with the complement having µ-measure zero. The polynomials φ n (z )= κ n z n + ..., orthonormal on the unit circle with respect to µ are uniquely determined by the requirement that κ n > 0 and  T φ n (ζ ) φ m (ζ ) dµ = δ n,m , n, m =0, 1,..., ζ ∈ T. The monic orthogonal polynomials Φ n are Φ n (z )= κ -1 n φ n (z )= z n + ..., and the values a n = a n (µ) def =Φ n (0) are known as the reflection coefficients. Let us recall that the orthogonal polynomials (both monic and orthonormal) as well as the measure itself are completely determined by their reflection coefficients through the Szeg˝ o recurrences (1) Φ n (z )= z Φ n-1 (z )+ a n Φ * n-1 (z ), n ∈ N def = {1, 2,...}, Φ 0 =1 (cf. [5, formula (11.4.7), p. 293]), and the connection between the reflection coeffi- cients and the leading coefficients κ n is given by (2) κ 2 n = n  k=1 ( 1 −|a n | 2 ) -1 , n ∈ N,κ 0 =1 (cf. [1, formula (8.6), p. 156]). Here the reversed * -polynomial of a polynomial p n of degree n is defined by p * n (z ) def = z n p n (1/ ¯ z). Moreover, each sequence a n of points from the open unit disk D comes up as a sequence of reflection coefficients for a Received by the editors December 13, 2001 and, in revised form, January 14, 2002. 2000 Mathematics Subject Classification. Primary 42C05. Key words and phrases. Measures on the unit circle, orthogonal polynomials, Szeg˝o recurrence relations. This material is based on work supported by the INTAS Grant 2000-272. c 2002 American Mathematical Society 1771 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use