Quadrature of a 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...

A TAXICAB VERSION OF A TRIANGLE' S APOLLONIUS CIRCLE

One of the most famous problems of classical geometry is the Apollonius' problem asks construction of a circle which is tangent to three given objects. These objects are usually taken as points, lines, and circles. This well known problem was posed by Apollonius of Perga ( about 262 - 190 B.C.) who was a Greek mathematician known as the great geometer of ancient times after Euclid and Archimede...

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