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Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are positively supported probability distribution functions such that $\frac{X_n}{n}$ converges in distribution to $X$ if $X_n\sim g_n$ and $X\sim g$.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are positively supported probability distribution functions such that $\frac{X_n}{n}$ converges in distribution to $X$ if $X_n\sim g_n$ and $X\sim g$.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, obviously with the inverted assumption that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are probability distribution functions.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are positively supported probability distribution functions such that $\frac{X_n}{n}$ converges in distribution to $X$ if $X_n\sim g_n$ and $X\sim g$.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are probability distribution functions.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: if anyone even has a small hint it would be very helpful for me, in the end this is just a real analysis problem where no complicated tools are needed. As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.

Given the space $F$ defined by: $$F=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(z,x)g(z)dz, x>0\right\},$$ I want to prove that the subspace $E$ of $F$ defined by $E=\bigcap_{n\geq 1} E_n$ where $$E_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k), x>0\right\}$$ is dense in $F$, assuming that $\displaystyle \lim_{n\rightarrow\infty}\sum_{k\in\mathbb{N}} f\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ for all $f$ in $C^2(\mathbb{R}_+^2;\mathbb{R})$, and where $g_n$ and $g$ are probability distribution functions.

The problem essentially comes from the interchange between the first and second components in the nonlocal boundary condition. More precisely, had the problem been to show that $G=\bigcap_{n\geq 1} G_n$ where: $$G_n=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\sum_{k\in\mathbb{N}} f\left(x,\frac{k}{n}\right)g_n(k), x>0\right\}$$ were dense in: $$H=\left\{f\in C^2(\mathbb{R}_+^2;\mathbb{R}):f(x,0)=\int_\mathbb{R} f(x,z)g(z)dz, x>0\right\},$$ it would have sufficed to define the sequence of functions $f_n$ by: $$ f_n(x,y)=f(x,y)+\frac{y}{\sum_{j\in\mathbb{N}}\frac{j}{n}g_n(j)}\left(\int_\mathbb{R} f(x,z)g(z)dz-\sum_{j\in\mathbb{N}} f\left(x,\frac{j}{n}\right)g_n(j)\right)$$ to deduce that $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(x,\frac{k}{n}\right)g_n(k)=\int_\mathbb{R} f(x,z)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$.

Unfortunately, I have not been able to find a suitable sequence of functions to use in my initial problem where the components are interchanged in the boundary condition. Any ideas or references to literature would be greatly appreciated.

Update: As an example, the sequence of functions $f_n$ defined by: $$ f_n(x,y)=f(x,y)+\mathbb{1}_{\{y>0\}}\left(\int_\mathbb{R} f(z,y)g(z)dz-\sum_{j\in\mathbb{N}} f\left(\frac{j}{n},y\right)g_n(j)\right)$$ would verify $f_n(x,0)=f(x,0)$, $\displaystyle \sum_{k\in\mathbb{N}} f_n\left(\frac{k}{n},x\right)g_n(k)=\int_\mathbb{R} f(z,x)g(z)dz$ and $\lim_{n\rightarrow\infty}\|f_n-f\|=0$, but $f_n$ is discontinuous (I am looking for a function in $C^2(\mathbb{R}_+^2;\mathbb{R}))$.