Superoscillations
It is commonly believed that a signal of bandwidth w Hertz cannot oscillate at frequencies higher than 2w Hertz. Yet, given any fixed bandwidth, there exist finite energy signals that oscillate in an arbitrarily fast way over arbitrarily long time intervals.
Such oscillations have been called superoscillations, and the signals that contain them are called superoscillating signals. The articles in this section study several issues raised by the existence of superoscillating signals.
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 wave functions 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 wave functions.
For any fixed bandwidth there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients or superoscillations can only occur in signals which possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared L_2 norm) as a function of the superoscillation's frequency, number, and maximum derivative. It briefly discusses some of the implications of superoscillating signals, in reference to information theory and time-frequency analysis, for example. It shows, among other things, that the required energy grows exponentially with the number of superoscillations, and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations' period.
It has been found that differentiable functions can locally oscillate on length scales that are much smaller than the smallest wave length contained in their Fourier spectrum. This phenomenon has been called superoscillation. Here, we consider the case of superoscillations in quantum mechanical wave functions. We find that superoscillations in wave functions lead to unusual phenomena that are of measurement theoretic, thermodynamic and information theoretic interest.
For any fixed bandwidth, there are finite energy signals which oscillate arbitrarily fast on arbitrarily long, finite time intervals. This paper investigates such signals, and their implications in reference to the transmission of information through low bandwidth channels.