A Tight Converse to the Spectral Resolution Limit via Convex Programming

It is now well understood that convex programming can be used to estimate the frequency components of a spectrally sparse signal from $2m+1$ uniform temporal measurements. It is conjectured that a phase transition on the success of the total-variation regularization occurs when the distance between the spectral components of the signal to estimate crosses $1\mathrm{/}m$. We prove the necessity part of this conjecture by demonstrating that this regularization can fail whenever the spectral distance of the signal of interest is asymptotically equal to $1\mathrm{/}m$.