= Conjugate symmetry of trigonometric polynomial coefficients
{title2=$p_{-k}=\overline{p_k}$}
A <trigonometric polynomial> is real on the <unit circle> precisely when its finite <coefficients> have the displayed symmetry. To prove necessity, conjugate its values on the circle, use $\overline z=z^{-1}$, and compare <coefficients>. Multiplying a <coefficient> difference by $z^d$ gives an ordinary <polynomial> vanishing at infinitely many points, so it is zero. Symmetry also gives $p(z^{-1})=\overline{p(\overline z)}$ for every nonzero complex $z$.
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