= Periodic dilation annihilates low Fourier modes
{title2=$\int_{\mathbb T}f(nx)e^{imx}\,dx=0\quad(0<|m|<n)$}
If $f$ is integrable and $2\pi$-periodic and $n$ is a positive integer, then $\int_0^{2\pi}f(nx)e^{imx}\,dx=0$ for every integer $m$ with $0<|m|<n$. Split the integral into $n$ equal intervals and substitute $y=nx-2\pi j$ to obtain
$$
\frac1n\int_0^{2\pi}f(y)e^{imy/n}\,dy
\sum_{j=0}^{n-1}e^{2\pi imj/n}=0.
$$
The <root of unity> sum vanishes by the finite <geometric series> formula. The zero mode is $\int f$, so a mean-zero dilated function is orthogonal to all <trigonometric polynomials> of degree at most $n-1$.
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