Superoscillating Sequences Towards Approximation in S or S′-Type Spaces and Extrapolation

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 S ( R , C ) or in some of its subspaces, tempered distributions or also...

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Bibliographic Details
Published inThe Journal of fourier analysis and applications Vol. 25; no. 1; pp. 242 - 266
Main Authors Colombo, F., Struppa, D. C., Yger, A.
Format Journal Article
LanguageEnglish
Published New York Springer US 15.02.2019
Springer Nature B.V
Subjects
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Partial Differential Equations
Sequences
Signal,Image and Speech Processing
Subspaces
32A15
47B38
Tempered distributions
Band-limited signals
32A10
Schwartz space
Approximation by superoscillations
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Summary: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 S ( R , 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.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-018-9592-8