Multiple quadrature using highly oscillatory quadrature methods

There has been recent interest in alpha-dense curves. These curves on Rn+1 densify the space and in effect convert a multi-dimensional integral into a one-dimensional integral. Inevitably one problem (dimensionality) is replaced by another (high oscillation rate), and the proposal here is to use generalized quadrature methods to develop specific methods to efficiently tackle the resulting integrals. These methods allow the generation of high-order formulae with given oscillatory weights w(x) which are solutions of a linear differential operator Lw=0. Some modifications are needed to handle integrations along α-dense curves.