= Uniqueness of a trigonometric series outside a finite set
If the symmetric partial sums of $\sum_{r\in\mathbb Z}a_re^{irt}$ tend to zero at every point outside a finite set, then all coefficients vanish. Almost-everywhere convergence first gives $a_{\pm n}\to0$. The twice-integrated series $F(t)=a_0t^2/2-\sum_{r\ne0}a_re^{irt}/r^2$ is continuous. <Riemann summation of a convergent trigonometric series> makes its symmetric second derivative zero away from the exceptional points, so it is affine on each intervening interval. <Sinc-squared summation of a sequence tending to zero> makes its symmetric first-order quotient zero everywhere, removing all possible affine corners. The globally affine function differs from a periodic function by $a_0t^2/2$, forcing $a_0=0$; periodicity then makes it constant, and integration against each nonzero frequency gives $a_r=0$.
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