One Approach to the Computation of Asymptotics of Integrals of Rapidly Varying Functions

We consider integrals of the form
$$ I(x,h)=\frac{1}{(2\pi h)^{k/2}}\int_{\mathbb{R}^k} f\biggl(\frac{S(x,\theta)}{h}\,,x,\theta\biggr)\,d\theta, $$
where $h$ is a small positive parameter and $S(x,\theta)$ and $f(\tau,x,\theta)$ are smooth functions of variables $\tau\in\mathbb{R}$, $x\in\mathbb{R}^n$, and $\theta\in\mathbb{R}^k$; moreover, $S(x,\theta)$ is real-valued and $f(\tau,x,\theta)$ rapidly decays as $|\tau|\to\infty$. We suggest an approach to the computation of the asymptotics of such integrals as $h\to0$ with the use of the abstract stationary phase method.