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 $g(t)=\sum_{j=1}^k v_j e^{i\lambda_jt}$ with $\lambda_j\in\mathbb R$ 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 $$|g(1+\delta)|\le O(k)\exp(O(k\sqrt\delta))\|g\|_{L^2[-1,1]}, \qquad 0\le\delta\le1 .$$ This improves on the $\exp(O(k^2\log k\cdot\delta))$ 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.