  We prove Szeg\H{o}-type trace asymptotics for translation-invariant operators
on polygons. More precisely, consider a Fourier multiplier $A=\mathcal{F}^\ast
\sigma \mathcal{F}$ on $\mathsf{L}^2(\mathbb{R}^2)$ with a sufficiently
decaying, smooth symbol $\sigma:\mathbb{C}\to\mathbb{C}$. Let $P\subset
\mathbb{R}^2$ be the interior of a polygon and, for $L\geq 1$, define its
scaled version $P_L:=L\cdot P$. Then we study the spectral asymptotics for the
operator $A_{P_L}=\chi_{P_L}A\chi_{P_L}$, the spatial restriction of $A$ onto
$P_L$: for entire functions $h$ with $h(0)=0$ we provide a complete asymptotic
expansion of $\operatorname{tr}h(A_{P_L})$ as $L\to\infty$. These trace
asymptotics consist of three terms that reflect the geometry of the polygon. If
$P$ is replaced by a domain with smooth boundary, a complete asymptotic
expansion of the trace has been known for more than 30 years. However, for
polygons the formula for the constant order term in the asymptotics is new. In
particular, we show that each corner of the polygon produces an extra
contribution; as a consequence, the constant order term exhibits an anomaly
similar to the heat trace asymptotics for the Dirichlet Laplacian.
