One fundamental property of Fourier Series

[viczhang]

Suppose the functions f(t) and g(t) are periodic with periods P and Q, respectively. If the ratio P/Q of their periods is a rational number, show that the sum f(t)+g(t) is a period function.

How to prove this?

 

[Dickfore]

If their ratio is a rational number, it means it can be represented as:

$$\frac{P}{Q} = \frac{m}{n}, \; m, n \in \mathbb{Z}^{+}, \; \mathrm{GCD}(m, n) = 1$$

Now, consider an interval of length:

$$R = \frac{P}{m}*\mathrm{LCM}(m, n)$$

What can you say about $f(x + R)$ and $g(x + R)$?