It is commonly assumed that a signal bandlimited to 2 Hz cannot oscillate at frequencies higher than Hz. In fact, however, for any fixed bandwidth, there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals. These localized fast transients, called superoscillations, can only occur in signals that possess amplitudes of widely different scales. This paper investigates the required dynamical range and energy (squared 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 also 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. . His research interests range from classical sampling theory to quantum information theory, quantum gravity, and inflationary cosmology. Prof. Kempf is a member of the Institute for Quantum Computing and an affiliated member of the Perimeter Institute for Theoretical Physics.

It has been found that functions can oscillate locally much faster than their Fourier transform would suggest is possible -a phenomenon called superoscillation. Here, we consider the case of superoscillating wave functions in quantum mechanics. We find that they possess rather unusual properties which raise measurement theoretic, thermodynamic and information theoretic issues. We explicitly determine the wave functions with the most pronounced superoscillations, together with their scaling behavior. We also address the question how superoscillating wave functions could be produced.

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