Superoscillations with Optimal Numerical Stability

A bandlimited signal can oscillate at a rate faster than its bandlimit. This phenomenon, called “superoscillation”, has applications e.g. in superresolution and superdirectivity. The synthesis of superoscillations is a numerically difficult problem. We introduce time translation σ as a design parameter and give an explicit closed formula for the condition number of the matrix of the problem, as a function of σ. This enables us to determine the best possible condition number, which is several orders of magnitude better than otherwise achievable.