  In this paper a class of oscillatory integrals is interpreted as a limit of
Lebesgue integrals with Gaussian regularizers. The convergence of the
regularized integrals is shown with an improved version of iterative
integration by parts that generates additional decaying factors and hence leads
to better integrability properties. The general abstract results are then
applied to the Cauchy problem for the one dimensional time dependent
Schr\"odinger equation, where the solution is expressed for C^n-regular initial
conditions with polynomial growth at infinity via the Green's function as an
oscillatory integral.
