Document Type
Article
Publication Date
2-1-2017
Abstract
In the past 50 years, quantum physicists have discovered, and experimentally demonstrated, a phenomenon which they termed superoscillations. Aharonov and his collaborators showed that superoscillations naturally arise when dealing with weak values, a notion that provides a fundamentally different way to regard measurements in quantum physics. From a mathematical point of view, superoscillating functions are a superposition of small Fourier components with a bounded Fourier spectrum, which result, when appropriately summed, in a shift that can be arbitrarily large, and well outside the spectrum. Purpose of this work is twofold: on one hand we provide a self-contained survey of the existing literature, in order to offer a systematic mathematical approach to superoscillations; on the other hand, we obtain some new and unexpected results, by showing that superoscillating sequences can be seen of as solutions to a large class of convolution equations and can therefore be treated within the theory of Analytically Uniform spaces. In particular, we will also discuss the persistence of the superoscillatory behavior when superoscillating sequences are taken as initial values of the Schrödinger equation and other equations.
Recommended Citation
Y. Aharonov, F. Colombo, I. Sabadini, D.C. Struppa, J. Tollaksen, The Mathematics of Superoscillations, Memoirs of the AMS 247 (2017), 1174. https://doi.org/10.1090/memo/1174
Peer Reviewed
1
Copyright
American Mathematical Society
Comments
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Memoirs of the American Mathematical Society, volume 247, in 2017 following peer review. The definitive publisher-authenticated version is available online at DOI: 10.1090/memo/1174