Black holes, bandwidths and Beethoven

Abstract

It is usually believed that a function $\mathrm{\phi (t)}$ whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component ${\omega }_{\max }.$ This is, in fact, not the case, as Aharonov, Berry, and others drastically demonstrated with explicit counterexamples, so-called superoscillations. It has been claimed that even the recording of an entire Beethoven symphony can occur as part of a signal with a 1 Hz bandwidth. Bandlimited functions also occur as ultraviolet regularized fields. Their superoscillations have been suggested, for example, to resolve the trans-Planckian frequencies problem of black hole radiation. Here, we give an exact proof for generic superoscillations. Namely, we show that for every fixed bandwidth there exist functions that pass through any finite number of arbitrarily prespecified points. Further, we show that, in spite of the presence of superoscillations, the behavior of bandlimited functions can be characterized reliably, namely through an uncertainty relation: The standard deviation $\mathrm{\Delta T}$ of samples ${\mathrm{\phi (t}}_n)$ taken at the Nyquist rate obeys ${\text{ΔT⩾1/4ω}}_{\max }.$ This uncertainty relation generalizes to variable bandwidths. For ultraviolet regularized fields we identify the bandwidth as the in general spatially variable finite local density of degrees of freedom.