Oscillatory integral

Remarks

Example: $n = 1$, $\phi(x) = x$ and $\psi$ smooth with compact support. Then $I(\lambda) = \int e^{i\lambda x}\psi(x) \, dx$ is the Fourier transform of $\psi$ at $-\lambda/(2\pi)$, and repeated integration by parts gives $\abs{I(\lambda)} \le C_N \abs{\lambda}^{-N}$ for every $N$. More generally, if $\nabla\phi \ne 0$ on the support of $\psi$, then $I(\lambda)$ decays faster than any power of $\lambda$.

Non-example: $\phi(x) = x$ and $\psi \equiv 1$ on $\R$. The amplitude does not have compact support, and $\int_\R e^{i\lambda x} \, dx$ does not converge as a Lebesgue integral. Such integrals are still given a meaning as distributions, which is the setting of the more general oscillatory integrals of Hörmander.

If $x_0$ is the only critical point in the support of $\psi$ and the Hessian of $\phi$ at $x_0$ is invertible, the method of stationary phase gives $I(\lambda) = O(\lambda^{-n/2})$ as $\lambda \to \infty$.