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

Abstract

We present a relation between rational Gauss-type quadrature formulas that approximate integrals of the form J μ ( F ) = ∫ − 1 1 F ( x ) d μ ( x ) , and rational Szegő quadrature formulas that approximate integrals of the form I μ ˚ ( F ) = ∫ − π π F ( e i θ ) d μ ˚ ( θ ) . The measures μ and μ ˚ are assumed to be positive bounded Borel measures on the interval [ − 1 , 1 ] and the complex unit circle respectively, and are related by μ ˚ ′ ( θ ) = μ ′ ( cos θ ) | sin θ | . 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 J μ ( F ) . Finally, we include some illustrative numerical examples.