Black Holes, Bandwidths and Beethoven
Abstract
It is usually believed that a function whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component. This is in fact not the case, as Aharonov, Berry and others drastically demonstrated with explicit counter examples, so-called superoscillations. The claim is that even the recording of an entire Beethoven symphony can occur as part of a signal with 1Hz bandwidth. Superoscillations have been suggested to account e.g. for transplanckian frequencies of black hole radiation. Here, we give an exact proof for generic superoscillations. Namely, we show that for every fixed bandwidth there exist functions which pass through any finite number of arbitrarily prespecified points. Further, we show that the behavior of bandlimited functions can be reliably characterized through an uncertainty relation for the standard deviation of the signals' samples taken at the Nyquist rate. This uncertainty relation generalizes to time-varying bandwidths.Keywords
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