Black holes, bandwidths and Beethoven

Abstract

It is usually believed that a function φ(t) whose Fourier spectrum is bounded can vary at most as fast as its highest frequency component ω max . 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: The standard deviation ∆T of samples φ(t n ) taken at the Nyquist rate obeys: ∆T ≥ 1/4ω max . This uncertainty relation generalizes to variable bandwidths. We identify the bandwidth as the in general spatially variable finite local density of the degrees of freedom of ultraviolet regularized fields.

Quantum averages of weak values

“…The exponential suppression of (5.7) near q + , where the weak value is maximal in magnitude, is generic of the phenomenon of Fourier superoscillations [40,41,42,43,47], exhibited by the amplitude n 2 ; j|e iĴzq |n 1 ; j near q + . This suppression imposes a "robustness" condition on the prior distribution in q if one is to measure eccentric weak values near q + : not only must the prior distribution be "sharp" around q = q + , but additionally it must show a sufficiently fast fall-off to overcome the exponential rise in likelihood.…”

What Is Variable Bandwidth?

“…Kempf and his collaborators [21,22,18] use a procedural concept of variable bandwidth, but (at least in the available literature) shy away from a formal definition. The parametrization of self-adjoint extensions of a differential operator leads to a class of algorithms that reconstruct or interpolate a function from certain samples.…”

Superoscillation in speckle patterns

“…This latter case has drawn much attention: at such places, the phase changes more rapidly than the constituent plane waves, hence the term "superoscillation" [1]. Superoscillatory waves, which locally vary much faster than their fastest Fourier component, have surprising and counterintuitive properties, and have recently been studied in a variety of systems, particularly signal processing, quantum mechanics, and optics [2][3][4][5]. Quantum mechanically, they fit into the general notion of weak measurements [6], and applications in optical imaging science have been suggested [7].…”