General theory of divergent, highly oscillatory integrals

Question

I'm interested in evaluation of integrals of the form $$\int e^{i S(x)}O(x) dx,$$ where $S(x)$ is a polynomial of degree higher than one and $O(x)$ is a polynomial or more generally, polynomialy bounded smooth function ($|O(x)|<C (1+x^2)^N$). The conventional way to provisionally define the value of such integral is to replace it by an expression of the form $$\int e^{i S(x)}O(x) \chi(\epsilon x) dx,$$ where $\chi$ is some smooth function such that $\chi(0)=1$, $\chi(x) \to 0$ as $x \to \infty$ sufficiently rapidly for the integral to converge and look at the limit $\epsilon \to 0$. This can be seen to work and give finite results in special cases which are computationally tractable. I want to know if some general results are known concerning the following questions: 1. Is the limit $\epsilon \to 0$ guaranteed to exist, 2. Is it independent of $\chi$, 3. Can a theory analogous to that of distributions be developed (I'd like to think of $e^{iS(x)}$ as sort of a distribution, but note that my $O(x)$ is much more general than test functions in Schwartz space).

Answer 1

Yes. The theory of Lagrangian distributions also known as Fourier Integral Operators.

See Hormander's papers Fourier Integral Operators I and II.

Or volume IV of his book.

Or Duistermaat's book on Fourier Integral Operators. (this is more readable.)