Optimal Extrapolation Bounds for Sparse Fourier Sums

We prove an optimal extrapolation theorem for $k$-sparse Fourier sums over arbitrary real frequencies, without any separation assumption, bounding how large such a sum can be just outside an interval on which its energy is observed. For every $g(t)=\sum_{j=1}^k v_j e^{iλ_jt}$ with $λ_j\in\mathbb R$ and every $x\ge1$, $$ |g(x)|\le k^{O(1)}\exp(O(k\mathop{\mathrm{arcosh}} x))\|g\|_{L^2[-1,1]} . $$ In the endpoint regime, this refines to the explicit bound $$ |g(1+δ)|\le O(k)\exp(O(k\sqrtδ))\|g\|_{L^2[-1,1]}, \qquad 0\leδ\le1 . $$ This improves on the $\exp(O(k^2\log k\cdotδ))$ growth estimate of Chen and Price (ICALP 2019), and the exponential scaling is optimal up to constants and polynomial factors in $k$. As an algorithmic consequence, we improve the cluster-center resolution of Chen--Price's clustered-frequency recovery algorithm by a factor of $k$, while preserving its sample complexity up to logarithmic factors. We also obtain exterior leverage-score and transfer bounds for sparse Fourier feature spaces, converting in-domain active-regression guarantees into essentially sharp prediction guarantees just outside the sampling interval.