On an uncertainty result by Donoho and Stark

In the work of Donoho and Stark, they study a manifestation of the uncertainty principle in signal recovery. They conjecture that, for a function with support of bounded size T, the maximum concentration of its Fourier transform in the low frequencies [-W/2,W/2] is achieved when the support of the function is an interval. They are able to prove a positive result under the extra assumption that WT $\leq$ 0.8, using an inequality with symmetric rearrangements. In our work, we present a more elementary proof of their result, while also relaxing the required bound to WT $\leq$ 1. Finally, we also study a discrete version of the problem, by considering complex polynomials and their concentration on subsets of the unit circle, and we prove an analogous problem. Lastly, this result is used to improve an inequality by Montgomery.