Exponential estimates for oscillatory integrals with degenerate phase functions

Abstract

In this paper we give precise asymptotic expansions and estimates of the remainder R(λ) for oscillatory integrals with non Morse phase functions, having degeneracies of any order k ⩾ 2. We provide an algorithm for writing down explicitly the coefficients of the asymptotic expansion analysing precisely the combinatorial behaviour of the coefficients (Gevrey type) and deriving optimal exponential decay estimates for the remainder when λ → ∞. We recapture the fundamental asymptotic expansions by Erdélyi (1956 Asymptotic Expansions (New York: Dover)). As it concerns the remainder estimates, it seems they are novel even for the classical cases. The main application of this machinery is a derivation of uniform estimates with respect to control parameters of celebrated oscillatory integrals in optics appearing in the calculations of the intensity of the light along the caustics (umbilics), see e.g. Arnold (1988 Singularities of Differentiable Maps vol II (Boston: Birkhäuser Boston Inc.)), (1974 USP. Mat. Nauk. 29 11–49) and Berry and Upstill (1980 Prog. Opt. 18 257–346). Finally, we mention that as an outcome of our abstract approach we obtain refinements for Morse phase functions provided suitable symmetry and Gevrey type regularity conditions on the phase functions and amplitudes hold. As far as we know, even this asymptotic expansion for the elliptic umbilic is a novelty.