A product of two functions is periodic; are the functions individually periodic?

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

$$P(x)=f(x)+g(x)$$

or a product

$$P(x)=f(x)g(x)$$

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 $$(m,n)\in\mathbb{Z}$$.

• Clearly not. Let $f(x)=\sin(x)+e^x$ and $g(x)=-e^x$ . Similarly for the product. – lulu Oct 4 '18 at 14:22
• 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 $f$ – F.Carette Oct 4 '18 at 14:23

1 Answer

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 $$g(x) = \frac{P(x)}{f(x)}$$.