Construction of Aharonov–Berry’s superoscillations

A simple method is described for constructing functions that superoscillate at an arbitrarily chosen wavelength scale. Our method is based on the technique of oversampled signal reconstruction. This allows us to explicitly demonstrate that the observed fragility of superoscillating wavefunctions is indeed mathematically closely connected to what in the communication theory community is known as the instability of oversampled signal reconstruction, confirming a previous conjecture. This is of potential interest, for example, concerning the understanding of the practical difficulties in experimentally producing superoscillatory wavefunctions. PACS numbers: 02.60.Ed, 02.30.Mv, 02.30.Nw, 03.65.Ta