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The Mathematics of Superoscillations


About this Title

Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa and Jeff Tollaksen

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 247, Number 1174
ISBNs: 978-1-4704-2324-7 (print); 978-1-4704-3709-1 (online)
DOI: https://doi.org/10.1090/memo/1174
Published electronically: February 1, 2017
Keywords:Superoscillations, convolution operators, infinite order differential operators, entire functions, Fourier transforms

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Physical motivations
  • Chapter 3. Basic mathematical properties of superoscillating sequences
  • Chapter 4. Function spaces of holomorphic functions with growth
  • Chapter 5. Schrödinger equation and superoscillations
  • Chapter 6. Superoscillating functions and convolution equations
  • Chapter 7. Superoscillating functions and operators
  • Chapter 8. Superoscillations in $SO(3)$

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.

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