I am currently studying the Stationary Phase Method, and I want to understand the proof given in the Geometric Asymptotics book by Guillemin and Sternberg, but I am experiencing some difficulties. I haven't used improper integrals for a long time and I am not so familiar with them, but I really need to understand this lemma (it is in page 5 of the book).
I take a function $h:\mathbb{R}^n \rightarrow \mathbb{R}$ such that $h, h'$ and $h''$ are bounded . My aim is to show that the integral $\int_{-\infty}^{+\infty} e^{-\lambda^2u}h(u)du$ exists, for $|\lambda|\geq1$ and $Re \lambda > 0$.
As far as I understand in the book the strategy is to split the integral into three parts $\int_{-\infty}^{-R} e^{-\frac{\lambda}{2}u^2}h(u)du +\int_{-R}^{R} e^{-\frac{\lambda}{2}u^2}h(u)du +\int_{R}^{+\infty} e^{-\frac{\lambda}{2}u^2}h(u)du$ for some $R$ and to treat each one independently. The integral doesn't make any problem when we try to evaluate on $[-R,R]$, and the two other cases are totally symmetric, so I just have to deal with the integration on $[R,+\infty]$.
For that the idea is to take a look at the integral $\int_{R}^{S} e^{-\frac{\lambda}{2}u^2}h(u)du$ for $0\lt R \lt S$. Then, after two integration by parts, we get that this is equal to $-\lambda^{-2}e^{-\frac{\lambda}{2}u^2}[\lambda \frac{h(u)}{u} - \frac{1}{u}(\frac{h(u)}{u})']_R^S + \lambda^{-2} \int_R^S e^{-\frac{\lambda}{2}u^2} (\frac{1}{u}(\frac{h(u)}{u})') du$.
From here comes the part where I am not understanding, in the book they are looking at the limit as $R\rightarrow\infty$, but I don't see how it is relevant for our concern. I have the impression that point that once $R$ is fixed as big enough, we want to make $S$ go to $\infty$ to show that this admits a limit and thus the integral is well defined.
Furthermore, the application comes then with Fubini's Theorem to prove that $\int_{\mathbb{R}^n} e^{\frac{\lambda}{2} Q(z)} f(z) dz$, where $f$ is compact support.
To do so, we break the function $f$ as follows: $f(z) = f(0) + \sum_{i=0}^n z_if_i(z)$, using Taylor series expansion. But now, $f_i$ are not necessarily compact support. For the previous lemma to apply, I have to first show that $f_1, f'_1, f''_1$ are bounded with respect to $z_1$, but also some more tricky condition like $z_2 \rightarrow \int_\mathbb{R} e^{-\frac{\lambda}{2}z_1^2} f_1(z_1,...,z_n) dz_1$ is bounded as well as its first two derivatives. I don't really see any reason for this to happen.
I know this might be basic question for some of you, but this all looks very confusing for now and I am having the strong impression that I am not looking at the problem from the correct angle, so any help would be greatly appreciated.
Thanks a lot