Positive interpolatory quadrature formulas and para-orthogonal polynomials

• Authors: Adhemar Bultheel;Leyla Daruis;Pablo González-Vera
• Venue: Journal of Computational and Applied Mathematics - Special issue: Proceedings of the conference on orthogonal functions and related topics held in honor of Olav Njåstad
• Year: 2005

We establish a relation between quadrature formulas on the interval [-1, 1] that approximate integrals of the form Jµ(F) = ∫-11 F(x) µ(x)dx and Szegö quadrature formulas on the unit circle that approximate integrals of the form Iω(f) = ∫-ππ f(eiθ)ω(θ) dθ. The functions µ(x) and ω(θ) are assumed to be weight functions on [-1, 1] and [-π, π], respectively, and are related by ω(θ) = µ(cos θ) |sin θ|. It is well known that the nodes of Szegö formulas are the zeros of the so-called para-orthogonal polynomials Bn, (z, τ) = Φn(z) + τΦn*(z), |τ| = 1, Φn(z) and Φn*(z), being the orthogonal and reciprocal polynomials, respectively, with respect to the weight function ω(θ). Furthermore, for τ = ±1, we have recently obtained Gauss-type quadrature formulas on [-1, 1] (see Bultheel et al. J. Comput. Appl. Math. 132(1) (2000) 1). In this paper, making use of the para-orthogonal polynomials with τ ≠ ±1, a one-parameter family of interpolatory quadrature formulas with positive coefficients for Jµ (F) is obtained along with error expressions for analytic integrands. Finally, some illustrative numerical examples are also included.