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

We present a relation between rational Gauss-type quadrature formulas that approximate integrals of the form , and rational Szegő quadrature formulas that approximate integrals of the form . The measures μ and are assumed to be positive bounded Borel measures on the interval [−1,1] and the complex unit circle respectively, and are related by . 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.