This question has been motivated by the post making sense of distributions on the diagonal.

Let $T$ be a tempered distribution on $\mathbb{R}^2$ and $\eta$ be a given mollifier on $\mathbb{R}$. For $f \in \mathcal{S}(\mathbb{R})$, let \begin{equation} \eta_{f,\epsilon}(x,y):= \frac{1}{\epsilon}f\Bigl(\frac{x+y}{2}\Bigr) \eta\Bigl(\frac{x-y}{\epsilon}\Bigr) \end{equation} and assume that the limit \begin{equation} \lim\limits_{\epsilon \to 0^+} T(\eta_{f,\epsilon}) \end{equation} exists for all $f \in \mathcal{S}(\mathbb{R})$.

Then, my issue is

For any other mollifier $\phi$ on $\mathbb{R}$, do we have $\lim\limits_{\epsilon \to 0^+} T(\phi_{f,\epsilon})=\lim\limits_{\epsilon \to 0^+} T(\eta_{f,\epsilon})$?

For a Schwartz function $F \in \mathcal{S}(\mathbb{R}^2)$ understood as a tempered distribution, a direct computation yields that \begin{equation} \lim\limits_{\epsilon \to 0^+} F(\phi_{f,\epsilon})=\lim\limits_{\epsilon \to 0^+} F(\eta_{f,\epsilon}) = \int_{\mathbb{R}} F(x,x) f(x) dx \end{equation}

Also, we know that the tempered dsitribution $T$ above can be approximated by Schwartz functions in the weak$^*$ topology. That is, we can find a sequence $F_n$ in $\mathcal{S}(\mathbb{R}^2)$ such that \begin{equation} F_n(g) \to T(g) \text{ as } n \to \infty \end{equation} for each $g \in \mathcal{S}(\mathbb{R}^2)$.

Therefore, my issue is reduced to finding a sequence $F_n$ approximating above $T$ such that \begin{equation} \lim\limits_{\epsilon \to 0^+} \lim\limits_{n \to \infty} F_n(\eta_{f,\epsilon}) = \lim\limits_{n \to \infty} \lim\limits_{\epsilon \to 0^+} F_n(\eta_{f,\epsilon}) \end{equation} and \begin{equation} \lim\limits_{\epsilon \to 0^+} \lim\limits_{n \to \infty} F_n(\phi_{f,\epsilon}) = \lim\limits_{n \to \infty} \lim\limits_{\epsilon \to 0^+} F_n(\phi_{f,\epsilon}) \end{equation} at the same time, for each $f \in \mathcal{S}(\mathbb{R})$.

However, I cannot see how to justify interchange of limits as written above. I guess I need some sort of "uniformity" in approximating $T$ by $F_n$.

Could anyone please help me?