Quadrature of the circle

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 ...

Lecture 1: The Quadrature of the Hermeneutic Circle

The first lecture will present pictorial semiotics within the framework of general semiotic theory. It will construe semiotics as a particular point of view taken on everything which is human or, more generally, endowed with life, rather than simply the continuation of the mixed or separate doctrines due to Saussure and Peirce. The historical part will describe briefly the development of pictor...

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...

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...

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