fundamental period of sum of two periodic functions

Question

Is there some formula to find fundamental period of sum of two periodic functions both of whose fundamental period is known. If yes what is the proof and the formula

Answer 1

The sum of two periodic functions with different periods doesn't need to be periodic, as evidenced by the following example: The functions $$f(x):=\sin(2\pi x), \quad g(x):=\sin{2\pi x\over\sqrt{2}}$$ are periodic with fundamental periods $1$ and $\sqrt{2}$ respectively, but the function $f+g$ is no longer periodic. (Since such sums occur in many examples of mathematical physics, a theory of almost periodic functions has been created.)

When the fundamental periods $p$ and $q$ of $f$ and $g$ have a rational quotient ${p\over q}$ then the least common multiple $L:={\rm lcm}(p,q)$ is certainly a period of $f+g$, but it need not be the fundamental period. It may occur that $f+g$ contains symmetries which were not present in the individual functions $f$ and $g$. Consider the following example: $$f(x):=\cases{1\quad&$(0\leq x<1)$\cr 0&$(1\leq x<4)$\cr f(x-4)\quad&(all $x$)\cr},\qquad g(x):=\cases{1\quad&$(2\leq x<3)$\cr 0&$(3\leq x<6)$\cr g(x-4)\quad&(all $x$)\cr}\quad.$$ Then $f$ and $g$ both have fundamental period $4$, but $f+g$ has fundamental period $2$. This shows that there is no general formula of the desired kind.