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Superoscillations

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New Perspectives on Approximation and Sampling Theory

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

A band-limited function may oscillate faster than its maximum Fourier component, and it may do so over arbitrarily long intervals. The goal of this chapter is to discuss this phenomenon, which has been called “superoscillation”. Although the theoretical interest in superoscillating functions is relatively recent, a number of applications are already known (in quantum physics, superresolution, subwavelength imaging and antenna theory). This chapter gives a brief account of how superoscillations appeared and developed and discusses their cost and some of their implications.

Dedicated to Paul L. Butzer, in friendship and high esteem.

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Notes

  1. 1.

    Paul Butzer has contributed significantly to sampling itself and to the study of its history. His work along the latter line includes [10, 14] and the more recent articles [11–13]. The history of the sampling principle is particularly challenging because the main ideas appeared in several independent works across the world: the USA, Europe, Japan [13] and Russia (see the translation of Kotel’nikov’s work in [3]). The multiple discovery of the sampling principle is the subject of [24].

  2. 2.

    See [35, 36], in which the so-called Nyquist rate is shown to be the minimum rate at which stable reconstruction can be performed.

  3. 3.

    As an example, Voelcker and Requicha [48] mention that speech that has been SSB-clipped (SSB modulated, clipped, retranslated to baseband) was known to possess much higher quality than LP-clipped speech. The references can be traced back to an information theory meeting held in 1956.

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Authors and Affiliations

  1. Universidade de Aveiro, Campus Universitário de Santiago, 3810-193, Aveiro, Portugal

    Paulo J. S. G. Ferreira

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  1. Paulo J. S. G. Ferreira
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Correspondence to Paulo J. S. G. Ferreira .

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  1. Department of Mathematical Sciences, DePaul University, Chicago, Illinois, USA

    Ahmed I. Zayed

  2. Department of Mathematics, University of Erlangen-Nuremberg, Erlangen, Germany

    Gerhard Schmeisser

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Ferreira, P.J.S.G. (2014). Superoscillations. In: Zayed, A., Schmeisser, G. (eds) New Perspectives on Approximation and Sampling Theory. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-08801-3_10

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