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When controlling an oscillatory integral, bump functions and bounded phase corrections are not very important

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Quick description

This is not a trick so much as a heuristic that allows one to not get too distracted with the unimportant components of an oscillatory integral (such as amplitude), and instead focus on the important components (namely, the geometry of the oscillation).

When controlling an oscillatory integral such as \int_{\R^d} e^{i \phi(x)} a(x)\ dx, the exact choice of amplitude function a(x) is rarely significant, so long as it is uniformly smooth with respect to all parameters of interest; one can often replace this amplitude function with another amplitude function b(x) so long as a and b have similar behaviour at the stationary points x_0 of \phi (e.g. a(x_0) = b(x_0)).

A corollary of this is that any smooth perturbation of the phase \phi(x) by a bounded correction \psi(x), thus replacing \int_{\R^d} e^{i \phi(x)} a(x)\ dx with \int_{\R^d} e^{i \phi(x) + i \psi(x)} a(x)\ dx should lead to an extremely similar integral. Thus, as a first approximation, any component of the phase \phi(x) which is both smooth and bounded can (heuristically, at least) be dropped for the purposes of trying to predict what the behavior of this integral should be.

Prerequisites

Harmonic analysis

Example 1

Suppose one wishes to estimate the integral \int_\R e^{i\lambda x^2} a(x)\ dx for some test function a \in C^\infty_c(\R) and some large parameter \lambda > 0. One can replace the cutoff function a(x) here by the more explicit function a(0) e^{-\pi x^2} (which agrees with a at the stationary point x=0), plus an error which vanishes at the stationary point. Computing the first term explicitly and bounding the second term via integration by parts, one obtains an asymptotic for this integral, namely a(0) (\frac{\pi}{-i\lambda})^{1/2} + O( \lambda^{-1} ); see "linearize the phase" for further discussion.

General discussion

Other examples of this general heuristic appear in "How to use the method of stationary phase to control oscillatory integrals".

See also "Getting rid of nasty cutoffs".