Two-frequency-dependent Gauss quadrature rules

We construct two-frequency-dependent Gauss quadrature rules which can be applied for approximating the integration of the product of two oscillatory functions with different frequencies β1 and β2 of the forms,yi(x)=fi,1(x) cos(βix)+fi,2(x) sin(βix), i=1,2,where the functions fi,j(x) are smooth. A regularization procedure is presented to avoid the singularity of the Jacobian matrix of nonlinear system of equations which is induced as one frequency approaches the other frequency. We provide numerical results to compare the accuracy of the classical Gauss rule and one- and two-frequency-dependent rules.