Positive rational interpolatory quadrature formulas on the unit circle and the interval

摘要

We present a relation between rational Gauss-type quadrature formulas that approximate integrals of the form Jμ(F) =∫_(-1)~1, F(x)dμ(x), and rational Szegoe quadrature formulas that approximate integrals of the form Iμ(F) = ∫_(-π)~π F(e~θ) dμe(θ). The measures μ and μe are assumed to be positive bounded Borel measures on the interval [-1,1] and the complex unit circle respectively, and are related by μe'(θ) = μ'(cosθ)| sinθ |. Next, making use of the so-called para-orthogonal rational functions, we obtain a one-parameter family of rational interpolatory quadrature formulas with positive weights for Jμ(F). Finally, we include some illustrative numerical examples.