Quadrature rules for the integration of the product of two oscillatory functions with different frequencies Original Research Article

We construct quadrature rules for the efficient computation of the integral of a product of two oscillatory functions y1(x) and y2(x), where yi(x)=fi,1(x)cos(βix)+fi,2(x)sin(βix), i=1,2, and the functions fi,j(x) are smooth. The weights are evaluated by the exponential fitting technique of Ixaru [Comput. Phys. Comm. 105 (1997) 1–19], which is now extended to cover the case of two frequencies. We give a numerical illustration on how the new rules compare for accuracy with the one-frequency dependent rules and with the classical ones.