On the estimation of multi-dimensional integrals with strongly oscillating integrands

The estimation of integrals with rapidly oscillating integrands is difficult, as positive and negative contributions will cancel nearly completely. The effect becomes more pronounced as the number of dimensions of the region of integration is increased. The article shows how an integral over ann-dimensional region can be reduced to integrals over itsn?1 dimensional surfaces in such a manner that the oscillating character of the integrand is taken into account. The method can be interpreted as the repeated application of integration by parts along lines normal to the wave fronts which are determined by the integrand. The remaining integral is estimated by the second mean value theorem of integral calculus. The integrations by parts appear only indirectly in the form of an application of Gauss' integral theorem to a vector field which is determined by the integrand.