Quadrature Rules for Trigonometric Polynomials

In the year 1959 Turetzkii introduced quadrature rules with maximal even trigonometric degree of exactness. The applications of such quadrature rules are in numerical integration of 2 π $$2\pi $$ -periodic functions as well as in the other fields of applied mathematics and other sciences. In this chapter we give a survey of recent results on quadrature rules for trigonometric polynomials. First, we present results about the quadrature rules of Gaussian type with maximal odd and even trigonometric degree of exactness, which are connected with the orthogonality in the space of trigonometric polynomials of integer and semi-integer degree, respectively. Special attention is given to the optimal set of quadrature rules with odd number of nodes for trigonometric polynomials in the sense of Borges and to its connections to multiple orthogonal trigonometric polynomials of semi-integer degree. A further part is dedicated to the case where an even number of nodes is fixed in advance for all mentioned quadrature rules. Also, the concept of anti-Gaussian quadrature rules for trigonometric polynomials is considered, as well as the averaged Gaussian quadrature rule for trigonometric polynomials, where attention is restricted to the case of an even weight function on ( − π , π ) $$(-\pi ,\pi )$$ . Finally, in addition to the theoretical results, the numerical methods for construction of the mentioned quadrature rules and appropriate numerical examples are given.