I am looking for an integral transform of $f(x)=\exp(-\frac{x^2}{2\sigma^2})$ such that:
$$(Tf)(u)=\int_{x_1}^{x_2}\, K(x,u)f(x)\,dx\,\stackrel{??}{=}\,\frac{a}{g(u)}$$
with $g(u)$ having zeros depending on $\sigma$.
Anybody has a clue if that could be possible? Or any similar expression, where I have my initial $f$ transformed to a function with asymptotes depending on $\sigma$.
Thanks!
EDIT: it is also fine to have some final function that has a maximum depending on $\sigma$, instead of an asymptote.
EDIT 2: here is an example of integral transform that gives me a function with a maximum depending on $\sigma$, but it turns out to be quite broaden around the maximum. A more narrowed function is required.
$$\int_0^{\infty}\,\exp\left(-\frac{x^2}{2\sigma^2}\right)\,x\,\sin(ax)\,\,dx= \sqrt{\frac{\pi}{2}}\sigma^3a\exp\left(-\frac{a^2\sigma^2}{2}\right)$$
with a maximum at $a=1/\sigma$. (Here I am considering only $a>0$)