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Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 242–266 | Cite as

Superoscillating Sequences Towards Approximation in \({\mathscr {S}}\) or \({\mathscr {S}}'\)-Type Spaces and Extrapolation

  • F. Colombo
  • D. C. StruppaEmail author
  • A. Yger
Article

Abstract

Aharonov–Berry superoscillations are band-limited sequences of functions that happen to oscillate asymptotically faster than their fastest Fourier component. In this paper we analyze in what sense functions in the Schwartz space \(\mathscr {S}(\mathbb {R},\mathbb {C})\) or in some of its subspaces, tempered distributions or also ultra-distributions, could be approximated over compact sets or relatively compact open sets (depending on the context) by such superoscillating sequences. We also show how one can profit of the existence of such sequences in order to extrapolate band-limited signals with finite energy from a given segment of the real line.

Keywords

Approximation by superoscillations Schwartz space Tempered distributions Band-limited signals 

Mathematics Subject Classification

32A15 32A10 47B38 

Notes

Acknowledgements

The authors express their gratitude to Irene Sabadini, for her participation in the many discussions that have led to the preparation of this article.

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Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Schmid College of Science and TechnologyChapman UniversityOrangeUS
  3. 3.IMB, Université de BordeauxTalenceFrance

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