I'm interested in the converse of the question here: Period of the sum/product of two functions.

Instead of "given two periodic functions, $f(x)$, $g(x)$, what is the period of a sum $f(x)+g(x)$ or product $f(x)g(x)$?", I am curious about:

"Given a periodic function $P(x)$ with period $T$ that can be written as a sum


or a product


is it necessarily true that $f(x)$ and $g(x)$ are periodic?"

If they are, it's pretty clear to me their periods would satisfy the same relationships as in the original question. For instance, $f$ and $g$ must have periods $p$ and $q$ such that $mp=mq=T$ for some $(m,n)\in\mathbb{Z}$.

  • 2
    $\begingroup$ Clearly not. Let $f(x)=\sin(x)+e^x$ and $g(x)=-e^x$ . Similarly for the product. $\endgroup$ – lulu Oct 4 '18 at 14:22
  • $\begingroup$ Do you consider the null function as periodic? If you do, there is a trivial counter-exemple for product, as $P(x)=0*f(x)=0$ would be periodic, no matter $f$ $\endgroup$ – F.Carette Oct 4 '18 at 14:23

For the sum, find some periodic function $P$ and a non-periodic function $f$, and set $g(x) = P(x) - f(x)$.

For the product, find some periodic function $P$, and a non-periodic function $f$ which never equals $0$, and set $g(x) = \frac{P(x)}{f(x)}$.


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