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.
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.
