Orthogonality properties of linear combinations of orthogonal polynomials

Let {P_n} be a sequence of orthogonal polynomials with respect to the measured on the unit circle and letP_n=P_n+_j^=1l_njP_n–j fornl, where_n,j . It is shown that the sequence of linear combinations {P_n},n2l, is orthogonal with respect to a positive measured if and only ifd is a Bernstein-Szegö measure andd is the product of a unique trigonometric polynomial and the Bernstein-Szegö measured. Furthermore for a given sequence ofP_n's an algorithm for the calculation of the _n,j's is provided.Supported by Dirección General de Investigación Cientifica y Técnica (DGICYT) of Spain and Österreichischer Akademischer Austauschdienst of Austria with grant 4B/1995.Also supported by the Austrian Fonds zur Förderung der wissenschaftlichen Forschung, project-number P9267-PHY.