Black Holes, Bandwidths and Beethoven

TL;DR

This paper proves that bandlimited functions can pass through any finite set of points, demonstrating superoscillations, and shows that their behavior can still be reliably characterized via an uncertainty relation, with implications for quantum field theory.

Contribution

It provides a rigorous proof of the existence of superoscillations in bandlimited functions and extends the uncertainty relation to characterize their behavior.

Findings

Existence of functions passing through any finite points within fixed bandwidth.

Superoscillations do not prevent reliable behavior characterization.

Generalization of results to time-varying bandwidths.

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. It has been claimed that even the recording of an entire Beethoven symphony can occur as part of a signal with 1Hz bandwidth. Bandlimited functions also occur as ultraviolet regularized fields. Their superoscillations have been suggested, for example, to resolve the transplanckian 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 which 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. This also generalizes to time-varying bandwidths. In QFT, we identify the bandwidth as the in general spatially variable finite local density of degrees of freedom of ultraviolet regularized fields.