Let {α 1 , α 2 , . . . } be a sequence of real numbers outside the interval [−1, 1] and µ a positive bounded Borel measure on this interval. We introduce rational functions ϕ n (x) with poles {α 1 , . . . , α n } orthogonal on [−1, 1] and establish some ratio asymptotics for these orthogonal rational functions, i.e. we discuss the convergence of ϕ n+1 (x)/ϕ n (x) as n tends to infinity under certain assumptions on the measure and the location of the poles. From this we derive asymptotic formulas for the recurrence coefficients in the three term recurrence relation satisfied by the orthonormal functions.

No abstract

No abstract

In this paper we consider a general sequence of orthogonal Laurent polynomials on the unit circle and we first study the equivalences between recurrences for such families and Szegő's recursion and the structure of the matrix representation for the multiplication operator in Λ when a general sequence of orthogonal Laurent polynomials on the unit circle is considered. Secondly, we analyze the computation of the nodes of the Szegő quadrature formulas by using Hessenberg and five-diagonal matrices. Numerical examples concerning the family of Rogers-Szegő q-polynomials are also analyzed.

Abstract. We consider the convergence of Gauss-type quadrature formulas for the integral ∞ 0 f (x)ω(x)dx, where ω is a weight function on the half line [0, ∞). The n-point Gauss-type quadrature formulas are constructed such that they are exact in the set of Laurent polynomialsIt is proved that under certain Carleman-type conditions for the weight and when p(n) or q(n) goes to ∞, then convergence holds for all functions f for which fω is integrable on [0, ∞). Some numerical experiments compare the convergence of these quadrature formulas with the convergence of the classical Gauss quadrature formulas for the half line.