Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
9509667 | Journal of Computational and Applied Mathematics | 2005 | 23 Pages |
Abstract
We establish a relation between quadrature formulas on the interval [-1,1] that approximate integrals of the form Jμ(F)=â«-11F(x)μ(x)dx and SzegÅ quadrature formulas on the unit circle that approximate integrals of the form IÏ(f)=â«-ÏÏf(eiθ)Ï(θ)dθ. The functions μ(x) and Ï(θ) are assumed to be weight functions on [-1,1] and [-Ï,Ï], respectively, and are related by Ï(θ)=μ(cosθ)|sinθ|. It is well known that the nodes of SzegÅ formulas are the zeros of the so-called para-orthogonal polynomials Bn(z,Ï)=Φn(z)+ÏΦn*(z), |Ï|=1, Φn(z) and Φn*(z), being the orthogonal and reciprocal polynomials, respectively, with respect to the weight function Ï(θ). Furthermore, for Ï=±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 Ïâ ±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.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
Adhemar Bultheel, Leyla Daruis, Pablo González-Vera,