Restricted Isometry for Variable-Density Continuous Frequency Sampling for Off-the-Grid Sparse Signals

Restricted isometry-type guarantees for continuous random sampling of Fourier frequencies in off-the-grid sparse models observed through a known point-spread function (PSF) are established. Specifically, for spike trains convolved with a fixed kernel and sampled on a finite bandwidth according to an arbitrary probability density, it is proven that the empirical frequency-domain energy concentrates around its expectation with near-linear sample complexity in the sparsity level. The constants are governed by two simple summaries of the PSF and the sampling density: an in-band energy term and a bounded-variation penalty that captures the effect of spike separation. Our analysis yields a practical design principle for variable-density sampling and admits a convex discretization. Numerical experiments illustrate tangible gains in localization when the sampling density is optimized accordingly.