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While digging in MSE I found the following

Linear combination of periodic functions need not to be periodic. If $f_1, f_2, f_3, \dots, f_k$ are periodic functions with fundamental periods $T_1, T_2, T_3, \dots, T_k$ respectively, then $\sum _{i=1} ^k a_i f_i$ where $a_1, a_2, \dots, a_k$ are constants is periodic if $\text{lcm } (T_1, T_2, \dots, T_k)$ exists.

Does anyone has a proof available or a link to it?

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  • $\begingroup$ Just to get an intuition, consider $\sin x + \{x\}$, where $\{\cdot\}$ is the fractional part function. This is clearly not periodic because the period of $\sin x$ is $2\pi$ which is irrational, while the period of $\{x\}$ is 1. $\endgroup$
    – abcd123
    Aug 16, 2020 at 16:23
  • $\begingroup$ I am aware of a proposition that a function of a sum of periodic functions is periodic if-f the ratio of the respective periods is rational , but somehow I got stumped in this one. $\endgroup$
    – Tolaso
    Aug 16, 2020 at 16:41

1 Answer 1

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Let $M = \mathrm{lcm}(T_1, ..., T_k)$. For each $i$, you have $T_i \text{ }| \text{ } M$, so because $f_i$ is $T_i$-periodic, it is also $M-$periodic.

So all the $a_i f_i$ are $M-$periodic, so it is not hard to see that there sum is also $M-$periodic.

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  • $\begingroup$ That would make sense $\endgroup$
    – Tolaso
    Aug 16, 2020 at 15:40

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