If $F(x)$ is a positive-definite function (possibly complex), is there any condition on $F$ that would ensure that $e{}^{\frac{+x^{2}}{2}}F(x)$ is also a positive-definite function?
Here, we assume that the function $e{}^{\frac{+x^{2}}{2}}F(x)$ is a $L{}_{2}$ function.
Such product is obviously not positive-definite in the general case but could we say that the product is positive-definite for a specific type/category of functions $F(x)$ or if $F(x)$ fulfil some specific conditions?
It would be helpful if someone could point me to a reference that address such questions. Thank you