Gaussian quadratures for oscillatory integrands

AbstractWe consider a Gaussian type quadrature rule for some classes of integrands involving highly oscillatory functions of the form f(x)=f1(x)sinζx+f2(x)cosζx, where f1(x) and f2(x) are smooth, ζ∈R. We find weights σν and nodes xν,ν=1,2,…,n, in a quadrature formula of the form ∫−11f(x)dx≈∑ν=1nσνf(xν) such that it is exact for all polynomials f1(x) and f2(x) from Pn−1. We solve the existence question, partially