TY - JOUR

T1 - The mathematics of superoscillations

AU - Aharonov, Yakir

AU - Colombo, Fabrizio

AU - Sabadini, Irene

AU - Struppa, Daniele C.

AU - Tollaksen, Jeff

N1 - Publisher Copyright:
©2017 by the American Mathematical Society. All rights reserved.

PY - 2017/5

Y1 - 2017/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85016445507&partnerID=8YFLogxK

U2 - 10.1090/memo/1174

DO - 10.1090/memo/1174

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AN - SCOPUS:85016445507

SN - 0065-9266

VL - 247

SP - 1

EP - 120

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

IS - 1174

ER -