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Representing Superoscillations and Narrow Gaussians with Elementary Functions

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Abstract

A simple addition to the collection of superoscillatory functions is constructed, in the form of a square-integrable sinc function which is band-limited yet in some intervals oscillates faster than its highest Fourier component. Two parameters enable tuning of the local frequency of the superoscillations and the length of the interval over which they occur. Away from the superoscillatory intervals, the function rises to exponentially large values. An integral transform generates other band-limited functions with arbitrarily narrow peaks that are locally Gaussian. In the (delicate) limit of zero width, these would be Dirac delta-functions, which by superposition could enable construction of band-limited functions with arbitrarily fine structure.

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Authors and Affiliations

  1. H.H. Wills Physics Laboratory, Tyndall Avenue, Bristol, BS81TL, UK

    M.V. Berry

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Correspondence to M.V. Berry.

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I thank Professors Fabrizio Colombo and Irene Sabadini for questions that stimulated this work, and the Centre for Disruptive Technologies of Nanyang Technological University, Singapore, and Macquarie University, Australia, for generous hospitality while this paper was written. My research is supported by the Leverhulme Trust.

Lecture given at the Seminario Matematico e Fisico di Milano on February 24, 2016

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Berry, M. Representing Superoscillations and Narrow Gaussians with Elementary Functions. Milan J. Math. 84, 217–230 (2016). https://doi.org/10.1007/s00032-016-0256-3

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  • DOI: https://doi.org/10.1007/s00032-016-0256-3

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