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

Positive trigonometric quadrature formulas and quadrature on the unit circle

We give several descriptions of positive quadrature formulas which are exact for trigonometric-, respectively, Laurent polynomials of degree less or equal to n − 1 − m, 0 ≤ m ≤ n − 1. A complete and simple description is obtained with the help of orthogonal polynomials on the unit circle. In particular it is shown that the nodes polynomial can be generated by a simple recurrence relation. As a ...

Error bounds for interpolatory quadrature rules on the unit circle

For the construction of an interpolatory integration rule on the unit circle T with n nodes by means of the Laurent polynomials as basis functions for the approximation, we have at our disposal two nonnegative integers pn and qn, pn + qn = n − 1, which determine the subspace of basis functions. The quadrature rule will integrate correctly any function from this subspace. In this paper upper bou...

Rational interpolation and quadrature on the interval and on the unit circle

Given a positive bounded Borel measure μ on the interval [−1, 1], we provide convergence results in Lμ2 -norm to a function f of its sequence of rational interpolating functions at the nodes of rational Gauss-type quadrature formulas associated with the measure μ. As an application, we construct rational interpolatory quadrature formulas for complex bounded measures σ on the interval, and give ...

A Connection Between Quadrature Formulas on the Unit Circle and the Interval

Quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

In this paper we investigate the Szegő-Radau and Szegő-Lobatto quadrature formulas on the unit circle. These are (n + m)-point formulas for which m nodes are fixed in advance, with m = 1 and m = 2 respectively, and which have a maximal domain of validity in the space of Laurent polynomials. That means that the free parameters (free nodes and positive weights) are chosen such that the quadrature...