Estimating exponential integrals with a smooth weight function (Q1337393)

Exponential integrals of the form \(I= \int_{-\infty}^ \infty g(x) e(f(x)) dx\) are considered. The assumptions on \(f\) and \(g\) are: \(g\) is supported in a finite interval \([-M,M]\), \(f\) and \(g\) have derivatives of any order, \(f^{(k)}(x) \ll FM^{-k}\), \(g^{(k)} (x)\ll GM^{-k}\) \((k=0,1, \dots)\), \(f(0)= f'(0) =0\), and \(f''(0)\asymp FM^{-2}\). Then, for any positive integer \(N\), the following formula holds: \(I= \sum_{n=0}^ N a_ n (f''(0) )^{-n-1/2}+ O(GM F^{-(N+ 2)/3})\), where the coefficients \(a_ n\) can be calculated in terms of the derivatives of \(f\) and \(g\) at zero.