On generalized quadrature rules for fast oscillatory integrals

In this paper, we present numerical analysis for the generalized quadrature rule for ∫abf(x)w(r,x)dx, where w(r,x)w(r,x) is an oscillatory function, and derive higher order generalized quadrature rules. The results show that the generalized quadrature rules are efficient for Bessel-trigonometric transformations and the accuracy increases when oscillation becomes faster.