I am trying to understand the stationary phase approximation as described on Wikipedia https://en.wikipedia.org/wiki/Stationary_phase_approximation. As necessary condition, they mention a compact support/exponential decline of the function g in the integral:
$\int_{R^n}g(x)e^{ikf(x)}dx$
However, in various textbooks, I was only able to find proofs, where compact support was explicitly required. In some more careless treatments, both conditions are nowhere to be found. So, if someone, who is familiar with the issue, could resolve this confusion for me and maybe argue why the particular condition has to be met, I'd be very grateful.