Superresolution via Sparsity Constraints

Abstract

Consider the problem of recovering a measure $\mu $ supported on a lattice of span $\Delta $, when measurements are only available concerning the Fourier Transform $\hat \mu (\omega )$ at frequencies $|\omega | \leqslant \Omega $. If $\Omega $ is much smaller than the Nyquist frequency ${\pi / \Delta }$ and the measurements are noisy, then, in general, stable recovery of $\mu $ is impossible. In this paper it is shown that if, in addition, we know that the measure $\mu $ satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length ${{4\pi } / \Omega } \cdot R$ there are at most R elements. Indeed, if the (unknown) support of $\mu $ is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like $\Delta ^{ - 2R - 1} $ as $\Delta \to 0$. This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astrono...