I am trying to prove that the set $D=\bigcap_{n\geq 1} \{f\in C^2 (\mathbb{R}) :f(0)=\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)\}$ is a core for the infinitesimal generator of reflecting brownian motion with jump boundary, i.e reflecting brownian motion which regenerates according to the density $g$ upon hitting $0$.
The generator of this process is given by $Af(x)=\frac{1}{2}f''(x)$ acting on the domain: $Dom(A)=\{f\in C^2 (\mathbb{R}) : f(0)=\int_{\mathbb{R}} f(u)g(u)du\}$.
By assumption, we are given the fact that $\lim_{n}\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)=\int_{\mathbb{R}} f(u)g(u)du$.
Using the definition of a core, I basically need to show that the closure of the set $\{(f,Af):f\in D\}$ is equal to the set $\{(f,Af):f\in Dom(A)\}$, i.e for any $f\in Dom(A)$, I can construct a sequence of functions $f_n$ in $D$ such that $\lim_{n} f_n=f$ and $\lim_{n} Af_n=Af$.
However, I am not sure how to go about finding such a sequence of functions. My initial idea was to consider $f_n (x) = f(x)+\sum_{k\geq 0} f(\frac{k}{\sqrt{n}})g_n (k)-\int_{\mathbb{R}} f(u)g(u)du$. In this case, we would obviously have $\lim_{n} f_n=f$ and $\lim_{n} Af_n=Af$, but the problem is $f_n$ is not necessarily in $D$. Any help or references to relevant literature would be greatly appreciated.