I did some calculations and I ended up with an integral
$$ I(t;\epsilon) = \frac{1}{\epsilon}\int_{\mathbb{R}} f(t') e^{-\frac{(t-t')^2}{4\epsilon^2}} dt' $$
where $f$ is in $L^2(\mathbb{R})$ and $\epsilon > 0$
I want to prove that as $\epsilon \to 0^+$ the integral converges to a multiple of $f(t)$ for fixed $t$. I've tried to use the Cauchy Schwartz inequality, but I ended up with nothing, except an upper bound indipendent from $\epsilon$.
Can you help me? I don't want to use the Fourier transform at this stage.
Assuming the integrand has a Gaussian, consider the equivalent integral by the interchange $t' \leftrightarrow t-t'$:
$$\frac{1}{\epsilon} \int_{\mathbb{R}} f(t-t') e^{-\frac{t'^2}{4\epsilon^2}} dt'$$
then use the substitution $s' = \frac{t'}{2\epsilon}$:
$$ = 2\int_{\mathbb{R}} f\left(t - 2\epsilon s'\right) e^{-s'^2}ds'$$
and we can use dominated convergence to get
$$\lim_{\epsilon\to 0^+} 2\int_{\mathbb{R}} f(t-2\epsilon s') e^{-s'^2}ds' = 2f(t)\int_{\mathbb{R}} e^{-s'^2}ds' = \sqrt{4\pi}f(t)$$
I will leave the finding of the dominating function to you.