... is the result in $L_{p}$?
A remark in my notes says yes but I can't see how to verify it.
As was pointed out to me in a previous question I asked last night, I need to show that the following integral is finite:
$\int_{-\pi}^{\pi}|\int_{-\pi}^{\pi}f(t-s)\phi_{n}(s)ds|^{p}dt < \infty$.
One of the properties of a summability kernel is that there exists a $C > 0$ such that $\int_{-\pi}^{\pi}|\phi_{n}(t)|dt\leq C$ for every $n\geq 1$. I feel like this could help if I could get $\phi$ by itself