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 J"@m(F)=@!"-"1^1F(x)@m(x)dx and Szego quadrature formulas on the unit circle that approximate integrals of the form I"@w(f)=@!"-"@p^@pf(e^i^@q)@w(@q)d@q. The functions @m(x) and @w(@q) are assumed to be weight functions on [-1,1] and [-@p,@p], respectively, and are related by @w(@q)=@m(cos@q)|sin@q|. It is well known that the nodes of Szego formulas are the zeros of the so-called para-orthogonal polynomials B"n(z,@t)=@F"n(z)+@t@F"n^*(z), |@t|=1, @F"n(z) and @F"n^*(z), being the orthogonal and reciprocal polynomials, respectively, with respect to the weight function @w(@q). Furthermore, for @t=+/-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 @t+/-1, a one-parameter family of interpolatory quadrature formulas with positive coefficients for J"@m(F) is obtained along with error expressions for analytic integrands. Finally, some illustrative numerical examples are also included.