Reconstruction of Signals: Uniqueness and Stable Sampling

Alexander Olevskii, Alexander Ulanovskii

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review

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 complete, or constitutes a frame in the space L2 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.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherBirkhauser
Pages9-49
Number of pages41
DOIs
StatePublished - 2020

Publication series

NameApplied and Numerical Harmonic Analysis
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017

Keywords

  • Exponential system
  • Frame
  • Interpolation set
  • Sampling set
  • Uniform densities
  • Uniqueness set
  • Universal Sampling
  • Universal completeness

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