I'm looking for an integral transform that maps $\cosh(x)$ into $\exp(x^2/2)$. More specifically I'm looking for a kernel $K(y,x)$ and a domain $\Omega \subset \mathbb{R}$ such that
$$ \int_\Omega dx \, K(y,x) \cosh(x) = \exp \left ( {\frac{y^2}{2}} \right ). $$
It would be nice if the kernel was somehow a "simple" well behaved function and invertible, such that there exist another kernel $G(x,y)$ and domain $\Gamma\subset \mathbb{R}$ such that
$$ \int_\Gamma dy \, G(x,y) \exp \left ( {\frac{y^2}{2}} \right ) = \cosh(x). $$
The Maclaurin series for $\cosh(x)$ and $\exp(x^2/2)$ both only involve even powers of $x$, which seems to suggest that such integral transform should exist. A somehow natural choice for $K(y,x)$ is
$$ K(y,x) = \frac{e^{-x/y}}{y} f(x/y) $$
but I haven't been able to find a suitable $f$ that does the job, so this may be a dead end.