= Lower-frequency Lp approximation to a single harmonic
{title2=$\operatorname*{argmin}_{t\in\mathcal T_{n-1}}\|a\cos nx+b\sin nx-t\|_p=0$}
For $1<p<\infty$ and $n\ge1$, 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.
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