Black holes, bandwidths and Beethoven (Q2737936)

The author gives an exact proof for generic superoscillations; namely, it is shown for every fixed bandwidth there exists functions that pass through any finite number of arbitrarily prespecified points. It is shown that among the function cutoff \(\omega_{\max}\) there always exist functions that pass through any finite number of arbitrarily prespecified points. Second, it is given a reliable characterization of the effect of frequency limitations on the behavior of the functions in terms of an uncertainty relation. Finally, it is shown that the characterization of frequency limited functions in terms of an uncertainty relation indeed allows a generalization to time varying bandlimits \(w_{\max}(t)\) for which superoscillations do not pose consistency problems.NEWLINENEWLINENEWLINETranslated into the language of field theory, the result will show, for example, that the number of degrees of freedom per unit volume is literally finite for ultraviolet cut off fields and that and how the density of degrees of freedom may, in general, be spatially varying.NEWLINENEWLINENEWLINEMost of this investigation is formulated in the language of communication theory, but these results apply wherever functions with a bounded Fourier spectrum occur, for example, in the case of field theories with an ultraviolet cutoff.