Quadrature Formulas of Gaussian Type for Fast Summation of Trigonometric Series

A summation/integration method for fast summing trigonometric series is presented. The basic idea in this method is to transform the series to an integral with respect to some weight function on $\RR_+$ and then to approximate such an integral by the appropriate quadrature formulas of Gaussian type. The construction of these quadrature rules, as well as  the corresponding orthogonal polynomials on $\RR_+$, are also considered. Finally, in order to illustrate the efficiency of the presented  summation/integration method two numerical examples are included.