Periodic Function as a Convolution

Question

I am reading a book about Signals and there is an exercise asking: Show that any periodic function can be written as the convolution of a non-periodic function with a train of Dirac delta distributions. Train here meaning a sum of the form: $$ \sum_{n=-\infty}^\infty c_n \delta(x-na) $$

Answer 1

Let $f$ be a periodic function of period $a>0$.

Now, consider $g$ to be the function that coincides with $f$ on $[0, a)$, and is $0$ anywhere else. $g$ is non-periodic, unless $f=0$. Then you can verify that for all $x$, $$f(x)=\sum_{n\in\mathbb Z}g(x-na)=\left (g \ast \sum_{n\in\mathbb Z}\delta_{na}\right)(x)$$