Orthogonal polynomials with respect to the sum of an arbitrary measure and a Bernstein–Szegö measure

Abstract

In the present paper we study the orthogonal polynomials with respect to a measure which is the sum of a finite positive Borel measure on [0, 2π] and a Bernstein-Szegö measure. We prove that the measure sum belongs to the Szegö class and we obtain several properties about the norms of the orthogonal polynomials, as well as, about the coefficients of the expression which relates the new orthogonal polynomials with the Bernstein-Szegö polynomials. When the Bernstein-Szegö measure corresponds to a polynomial of degree one, we give a nice explicit algebraic expression for the new orthogonal polynomials.

Basic hypergeometric functions and orthogonal Laurent polynomials

“…Since then, these polynomials which bear the name of Szegő were extensively studied by many. We cite, for example, [5], [6], [7], [10], [19], [20], [22], [25] and [27] as some of the very recent contributions. The recent publications of the two excellent volumes [23] and [24] by Simon have given a boost to the interest in studying these polynomials.…”

Szegő polynomials from hypergeometric functions

“…Since then, these polynomials which bear the name of Szegő have been extensively studied by many. We cite, for example, [3], [4], [5], [8], [14], [15], [16] and [19] as some of the very recent contributions. The recent publications of the two excellent volumes [17] and [18] by Simon have given, apart from a thorough perspective of the subject, a new boost to the interest in these polynomials.…”