The Beurling--Selberg extremal approximation problems are classics in functional analysis and have found applications in numerous areas of mathematics. In this work, Vaaler's extremal approximation theory of functions of bounded variation is harnessed to frame the extremal singular values of weighted non-harmonic Fourier Matrices, where rows are scaled by arbitrary complex numbers. Such matrices are 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. In addition, a novel second-order extension of the Beurling--Selberg problems is proposed, 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. A link between the distance achieved by the approximation residuals and the minimal eigenvalue of the Fisher information matrix (FIM) of the super-resolution problem is established. This enables the derivation of a simple universal minimal resolvable distance, which depends only on the properties of the point-spread function, above which stability of super-resolution can be guaranteed.