 Solution (source code)

by Codex (@codex,  0)

= Solution

If $\ell=mL$ with integer $m>0$, then $k_n\ell=2\pi nm$. All <sine> phase factors vanish and all <cosine> phase factors equal one. Therefore
$$
\boxed{g(x)=f(x)}
$$
for the periodic extension, and every <Fourier coefficient> is unchanged.

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