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Erschienen in:
Claude Shannon: American Genius Kapitel lesen Erstes Kapitel lesen

2020 | OriginalPaper | Buchkapitel

Reconstruction of Signals: Uniqueness and Stable Sampling

verfasst von : Alexander Olevskii, Alexander Ulanovskii

Erschienen in: Sampling: Theory and Applications

Verlag: Springer International Publishing

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Abstract

The classical sampling problem is to reconstruct a continuous signal with a given spectrum S from its samples on a discrete set Λ. Through the Fourier transform, the problems ask for which sets of frequencies Λ is the exponential system
$$\displaystyle \{e^{i\lambda t},\, \lambda \in \varLambda \} $$
complete, or constitutes a frame in the space L 2 on a given set S of finite measure? When S is a single interval, these problems were essentially solved by A. Beurling, A. Beurling and P. Malliavin in terms of appropriate densities of the discrete set Λ. H. Landau extended the necessity of the density conditions in these results to the general bounded spectra. However, when S is a disconnected set, no sharp sufficient condition for sampling and completeness can be expressed in terms of the density of the set Λ. Not only the size, but also the arithmetic structure of Λ comes into the play. This paper gives a short introduction into the subject of sampling and related problems. We present both classical and recent result.

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Vorheriges Kapitel Claude Shannon: American Genius
Nächstes Kapitel Sampling Theory in a Fourier Algebra Setting
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Metadaten
Titel
Reconstruction of Signals: Uniqueness and Stable Sampling
verfasst von
Alexander Olevskii
Alexander Ulanovskii
Copyright-Jahr
2020
Buch
Sampling: Theory and Applications

Print ISBN: 978-3-030-36290-4

Electronic ISBN: 978-3-030-36291-1

Copyright-Jahr: 2020

DOI
https://doi.org/10.1007/978-3-030-36291-1_2