Suppression of superoscillations by noise

Bandlimited functions can vary faster than their highest Fourier component. Such 'superoscillations' result from near-perfect destructive interference among the Fourier components and correspond to large values of the phase gradient (local wavenumber). Superoscillations that are strong and extend over a large interval occur where functions are exponentially small. The associated interference is vulnerable to noise, in particular random phases. Averaging over the phases, modelled as independent Gaussian variables with a specified rms value, enables the suppression of superoscillations to be described quantitatively; very weak phase noise suffices. Strong noise generates functions that are essentially random, and the remaining well-understood superoscillations are localised in small intervals. The theory is illustrated by computations with an explicit superoscillatory function.