A Gauss quadrature rule for oscillatory integrands Original Research Article

We consider the Gauss formula for an integral, ∫−11y(x) dx≈∑k=1Nwky(xk), and introduce a procedure for calculating the weights wk and the abscissa points xk, k=1,2,…,N, such that the formula becomes best tuned to oscillatory functions of the form y(x)=f1(x)sin(ωx)+f2(x)cos(ωx) where f1(x) and f2(x) are smooth. The weights and the abscissas of the new formula depend on ω and, by the very construction, the formula is exact for any ω provided f1(x) and f2(x) are polynomials of class PN−1. Numerical illustrations are given for N between one and six.