Quadrature rules using first derivatives for oscillatory integrands

We consider the integral of a function y(x), I(y(x))=∫−11y(x) dx and its approximation by a quadrature rule of the formQN(y(x))=∑k=1N wky(xk)+∑k=1N αky′(xk),i.e., by a rule which uses the values of both y and its derivative at nodes of the quadrature rule. We examine the cases when the integrand is either a smooth function or an ω dependent function of the form y(x)=f1(x) sin(ωx)+f2(x) cos(ωx) with smoothly varying f1 and f2. In the latter case, the weights wk and αk are ω dependent. We establish some general properties of the weights and present some numerical illustrations.