Periodic dilation annihilates low Fourier modes (${\int }_{\mathbb{T}}f(nx)e^{imx}dx=0(0<|m|<n)$)

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 obtainThe 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 $n-1$.