Notes on infinite dimensional integrations of probabilistic and oscillatory type

where s is a “type parameter”, usually s = −i or s = 1, s = −i standing for “oscillatory”, s = 1 for “probabilistic type”. Φ is a real-valued function, in the probabilistic case assumed to be lower bounded; it is called “phase function”; f is a complex-valued function, called “amplitude function”; γ is thought to be a member of some space Γ “on which the integration extends”; dγ is a heuristic “flat measure”. If Γ is finite dimensional, say Γ = R, then dγ is thought of as Lebesgue measure. If Γ is infinite dimensional, dγ has, as it stands, no clear meaning, as we shall see in the following sections. Typical examples in finite dimensions are: for s = −i the oscillatory integrals ∫ Rn ef(γ)dγ for which there is a theory which does not require absolute convergence of the integral, still keeping certain basic properties of Daniel’s approach to integration (i.e. integrals as linear continuous functionals, see, e.g. [30]).This is part of the finite dimensional theory of oscillatory integrals discussed in connection with topics like Fourier integral operators, resolution