A Gauss quadrature rule for oscillatory integrands

We consider the Gauss formula for an integral, integral (1)(-1) y(x) dx approximate to Sigma (N)(k=1) w(k)y(x(k)) and introduce a procedure for calculating the weights w(k) and the abscissa points x(k), k = 1, 2,.... N, such that the formula becomes best tuned to oscillatory functions of the form y(x) = f(1) (x) sin(wx) + f(2)(x) cos(wx) where f(1)(x) and f(2)(x) are smooth. The weights and the abscissas of the new formula depend on omega and, by the very construction, the formula is exact for any omega provided f(1)(x) and f(2)(x) are polynomials of class PN-1. Numerical illustrations are given for N between one and six