Second-Order Beurling Approximations and Super-Resolution from Bandlimited Functions

The Beurling–Selberg extremal approximation problems are classics in functional analysis and have found applications in numerous areas of mathematics. Of particular interest, optimal solutions to those problems can be exploited to provide sharp bounds on the condition number of Vandermonde matrices with nodes on the unit circle, which is of great interest to many inverse problems, including super-resolution. However, those solutions have non-derivable Fourier transforms, which impedes their use in a stability analysis of the super-resolution problem. We propose novel second-order extensions to Beurling–Selberg problems, where the approximation residual to functions of bounded variation (BV) is constrained to faster decay rates in the asymptotic, ensuring the smoothness of their Fourier transforms. We harness the properties of those second-order approximants by establishing a link between the norms of the residuals and the minimal eigenvalue of the Fisher information matrix (FIM) of the super-resolution problem. This enables the derivation of a simple and computable minimal resolvable distance for the super-resolution problem, depending only on the properties of the point-spread function, above which stability can be guaranteed.