It is provable that $f\to f_\lambda\Rightarrow f_\lambda\to f\Rightarrow f_\lambda*g\to f*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that particular case, $f_\lambda*g\to f*g\Rightarrow f\to f_\lambda$f_\lambda\to f$?
In other words: It is easy to prove that the existence of $\lim_{\lambda\to\lambda_0}$ \lim_{\lambda\to\lambda_0}\langle f_\lambda(t),\phi(t)\rangle$ implies existence of $\lim_{\lambda\to\lambda_0}\langle f_\lambda(t),\int_{t-\triangle t}^{t+\triangle t}\phi(\tau)d\tau\rangle$ where $\phi(t)$ denotes test function, and $f$ can be a distribution (since $\int_{t-\triangle t}^{t+\triangle t}\phi(\tau)d\tau$ is a test function itself). I am wondering does the converse hold, i.e. does the existence of latter limit imply existence of the former (note that not all test functions are expressible as $\int_{t-\triangle t}^{t+\triangle t}\phi(\tau)d\tau$)t}\phi(\tau)d\tau$)?
It may be a silly question, but it would be of much use to know the answer: proof if it holds, or a counterexample if doesn't.
