This is a rather long problem and I'm having difficulties with a very specific part.
The question starts with a function $\phi\in L_{1}(\mathbb{R})$ which vanishes outside of $(-\pi,\pi)$.
Then for each $n\geq 1$, $\phi_{n}$ is defined to be the $2\pi$-periodic function by $\phi_{n}(t) = n\phi(nt)$ for $t\in (\frac{-\pi}{n}, \frac{\pi}{n})$ and $\phi_{n}(t) = 0$ for $t\in [-\pi,-\frac{\pi}{n})\cup(\frac{\pi}{n}, \pi)$.
The first step was to show that $\{\phi_{n}\}$ is a summability kernel (done).
Next I needed to show that if $f\in L_{1}(\mathbb{T})$, and if $\phi$ above happens to be in $C^{\infty}(\mathbb{R})$, then for each $n\geq 1$, $f*\phi_{n}\in C^{\infty}(\mathbb{R})$ (done) and $supp(f*\phi_{n}) \subset supp(f) + supp(\phi_{n})$.
That is if $(f*\phi_{n})(t)\neq 0$, then $t = t_{1} + t_{2}$ for some $t_{1}, t_{2}$, where $f(t_{1})\neq 0$ and $\phi_{n}(t_{2})\neq 0$.
It is this last part that I bothers me.
UPDATE:
This almost seems too easy to be true. So that's why I'm a little sheepish about it.
Let $t\in supp(f*\phi_{n})$. Since $f*\phi_{n}\neq 0$, we have $\int_{-\pi}^{\pi}f(t-s)\phi_{n}(s)ds \neq 0$. Thus f(t-s')\phi_{n}(s')\neq 0 for at least one s'\in[-\pi,\pi). Then t = (t-s')+ s' and t-s'\in supp(f) and s'\in supp(\phi_{n}). Thus $supp(f*\phi_{n}) \subset supp(f) + supp(\phi_{n})$.
Is this reasonable or am I oversimplifying the problem?