Let $g(u,\delta)=E[f(x)]$ where the expectation is over $N(u,\delta^2)$.
Is there a characterization what function $g(u,\delta)$ can be produced this way? Is there a procedure solve the inverse problem, that is, given 
$g(u,\delta)$, the procedure should output $f(x)$? 


This seems like a fundamental question. But I don't know any 
reference about it. Any reference will be appreciated.
Thanks a lot. 



What function is a Gaussian integral
I think we can give a quick characterization adding some smootheness and boundness hypothesis via Stochastic calculus. They are exactly the solutions to the heat equation.


First of all we have $$g(u,\delta)=E[f(u+\delta Z)]$$ where $Z$ is a standard gaussian so necessarily $g(u,0)=f(u)$ so the function $f$ is fully determined by $g$. In the following i will change the definition of $g$ slightly for convenience: I will define $g(u,t)=E[f(u+\sqrt t Z)]$ where $Z$ is an $N(0,1)$.
We can rewrite this as $g(u,t)=E[f(u+W\_t)]$ where $W\_t$ is a standard Brownian motion. Now if we suppose the function $f$ is $C^2$ then by Itô formula we have 
$$f(u+W\_t)=f(u)+\int\_0^t f^{'}(u+W\_s)dWs+\frac 12\int\_0^t f^{''}(u+W\_s)ds$$
and then $g(u,t)=f(u)+\frac12\int\_0^t E[f^{''}(u+W\_s)]ds$


taking the derivative wrt to $t$ gives $$\partial\_t g(u,t)=\frac12 E[f^{''}(u+W\_t)]=\frac12 \partial^2\_ug(u,t)$$
The function $g$ is hence a solution to the heat equation.


Conversely if $g$ verifies the heat equation we can represent it as an expectation via Feyman Kac for example. the idea is to consider $g(u+Ws, t-s)$ as a function of $s$ and apply Itô on it:
$$dg(u+W\_s,t-s)=\left( -\partial\_t g(u+W\_s,t-s)+\frac 12 \partial^2\_ug(u+W\_s,t-s) \right ) ds+\partial\_ug(u+W\_s,t-s)dW\_s$$
then integrating between $0$ and $t$
$$g(u+W\_t,0)=g(u,t)+\int\_0^t \partial\_ug(u+W\_s,t-s)dW\_s$$
taking the expectation gives finally
$$g(u,t)=E[g(u+W\_t,0)]$$


If you're not familiar with Itô calculus, you can get the PDE for $g$ by defining the kernal $K(t,x)=\frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t}}$ and noticing that it verifies the heat equation $\partial\_t K(t,x)=\frac 12 \partial^2\_xK(x,t)$
then $g(u,t)=\int\_R f(u+x)K(t,x)dx$. This propagates the heat equation of $K$ to $g$. And conversely if $g$ verifies the heat equations then we can write $g(u,t)$ as an integral of the boundary value at $t=0$ of $g$ against $K$ as $K$ may be seen as a Green function.


