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The Mathematics of Superoscillations

About this Title

Yakir Aharonov, Schmid College of Science and Technology, Chapman University, Orange 92866, California, Fabrizio Colombo, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy, Irene Sabadini, Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133 Milano, Italy, Daniele C. Struppa, Schmid College of Science and Technology, Chapman University, Orange 92866, California and Jeff Tollaksen, Schmid College of Science and Technology, Chapman University, Orange 92866, California

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)
Published electronically: February 1, 2017
Keywords: Superoscillations, convolution operators, infinite order differential operators, entire functions, Fourier transforms
MSC: Primary 42A16, 42A38, 81P15, 47B38, 35Q41

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Table of Contents


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


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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