A New Class of Orthogonal Polynomials1

A new class of orthogonal polynomials is introduced which generalizes the Bernstein-Szegö polynomials and includes the associated polynomials as well. The purpose of this paper is to give a natural extension of the Bernstein-Szegö orthogonal polynomials for a general class of weight functions. A nonnegative function w defined on the real line is called a weight function if w > 0, fRw > 0 and all the moments of w are finite. For a given weight w one can construct a unique system of polynomials {pn(w)}TM=0 such thatpn(w, x) = yn(w)x" + ' ' ' >' Y„(w) > O» and ÍRPn(w)Pm(w)w = &mnThe Bernstein-Szegö orthogonal polynomials [13, §2.6] are the ones corresponding to the weight w defined by (1) w(x) = (l-x)±1/2(\+x)±l/2/p(x), -1<JC.<1, with w supported in [-1,1] where p is a positive polynomial in [-1,1], and these polynomials play a fundamental role in Szegö's theory, in particular, they are used to solve Szegö's extremal problem, they are applied to obtain Bernstein-Szegö's asymptotics for orthogonal polynomials, and they make it possible to prove theorems about equiconvergence of orthogonal polynomial series and Fourier series [5-7 and 13]. If g g L*(R) then the Stieltjes transform S(g) of g is defined by