Superoscillations and leaky spectra

Superoscillatory functions—band-limited functions with local oscillations faster than their fastest Fourier components—are extended to families that are ‘leaky’—not band-limited—but possess the same local oscillations. Two different extensions are presented. For deterministic functions, the prototype superoscillatory function is embedded in a leaky one-parameter family that can be studied in detail analytically. For Gaussian random superoscillatory functions, the coefficients in the Fourier series that represents them are modified so as to change the power spectrum from band-limited to leaky. Fast oscillations in leaky functions can be indistinguishable from superoscillations, but in practice this is unlikely to be important.