It is known that

\begin{equation*} I(p) = \sqrt { \frac {4 \mathrm{i} p} {\pi}}\int \limits _{-\infty} ^{\infty} \mathrm{e}^{- \mathrm{i} p x^2} \varphi (x) \mathrm{d}x \end{equation*}

is a bounded smooth function on $(0,\infty)$ with $\lim \limits _{p \to \infty} I(p) = \varphi(0)$ ($\varphi$ having compact support).

Does one have an even better understanding of $I(p)$ as $p$ aproaches $\infty$? You see, I am not only interested in its limit, but also in $how$ it tends to it. Something like the "dominant" part of $I(p)$, or some tight bound (of it or its absolute value), or any other approximation would be of help.