Estimating the Frequency of a Clustered Signal

We consider the problem of locating a signal whose frequencies are "off grid" and clustered in a narrow band. Given noisy sample access to a function $g(t)$ with Fourier spectrum in a narrow range $[f_0 - \Delta, f_0 + \Delta]$, how accurately is it possible to identify $f_0$? We present generic conditions on $g$ that allow for efficient, accurate estimates of the frequency. We then show bounds on these conditions for $k$-Fourier-sparse signals that imply recovery of $f_0$ to within $\Delta + \tilde{O}(k^3)$ from samples on $[-1, 1]$. This improves upon the best previous bound of $O\big( \Delta + \tilde{O}(k^5) \big)^{1.5}$. We also show that no algorithm can do better than $\Delta + \tilde{O}(k^2)$. In the process we provide a new $\tilde{O}(k^3)$ bound on the ratio between the maximum and average value of continuous $k$-Fourier-sparse signals, which has independent application.