Positive interpolatory quadrature formulas and para-orthogonal polynomials

摘要

We establish a relation between quadrature formulas on the interval [-1,1] that approximate integrals of the form Jμ(F)=∫-11F(x)μ(x)dx and Szegő quadrature formulas on the unit circle that approximate integrals of the form Iω(f)=∫-ππf(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(cosθ)|sinθ|. It is well known that the nodes of Szegő 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.