Generalized Fresnel integrals

A general class of (finite dimensional) oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic expansion for “strong oscillations” is given. The expansion is in powers of ?1/2M, where ? is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions.