Positive interpolatory quadrature formulas and para-orthogonal polynomials

Abstract

We establish a relation between quadrature formulas on the interval [ - 1 , 1 ] that approximate integrals of the form J μ ( F ) = ∫ - 1 1 F ( x ) μ ( x ) d x and Szegő quadrature formulas on the unit circle that approximate integrals of the form I ω ( f ) = ∫ - π π f ( e i θ ) ω ( θ ) d θ . 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 Szegő formulas are the zeros of the so-called para-orthogonal polynomials B n ( z , τ ) = Φ n ( z ) + τ Φ n * ( z ) , | τ | = 1 , Φ n ( z ) and Φ n * ( 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 J μ ( F ) is obtained along with error expressions for analytic integrands. Finally, some illustrative numerical examples are also included.