Lower-frequency Lp approximation to a single harmonic (${\mathrm{argmin}}_{t\in {\mathcal{T}}_{n-1}}\Vert a\cos nx+b\sin nx-t{\Vert }_p=0$)

For $1<p<\infty$ and $n\ge 1$, existence follows from finite-dimensional compactness and uniqueness from strictly convex normed space geometry. Translation by $2\pi /n$ and negative translation by $\pi /n$ preserve the target and the error norm. Uniqueness forces the minimizing trigonometric polynomial to be invariant under the first operation and the negative of itself under the second. Invariance removes every nonzero frequency below $n$; negative invariance removes the remaining constant.