Universal sampling of band-limited signals

Alexander Olevskii; Alexander Ulanovskii

doi:10.1016/j.crma.2006.04.015

Harmonic Analysis

Universal sampling of band-limited signals
[Échantillonnage universel de signaux à spectre borné]

Présenté par : Jean-Pierre Kahane

Alexander Olevskii ¹ ; Alexander Ulanovskii ²

¹ School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
² Stavanger University, NO-4036 Stavanger, Norway

Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 927-931.

Résumé
Abstract

Nous posons le problème de l'existence d'ensembles discrets, de densité donnée, permettant par échantillonnage la reconstitution, ou la reconstitution stable, de tout signal dont le spectre a une mesure de Lebesgue assez petite. Pour la reconstitution stable, nous montrons que la réponse dépend de manière cruciale du fait que le spectre soit compact ou soit dense. La reconstitution simple, par contre, est toujours possible.

We ask if there exist universal sampling sets of given density, which provide reconstruction or stable reconstruction of every band-limited signal whose spectrum has a small Lebesgue measure. For the stable reconstruction, we show that it is crucial whether the spectrum is compact or dense. On the other hand, the non-stable universal reconstruction is possible in general situation.

Reçu le : 2006-03-29
Accepté le : 2006-03-30
Publié le : 2006-05-22

DOI : 10.1016/j.crma.2006.04.015

Affiliations des auteurs :

Alexander Olevskii ¹ ; Alexander Ulanovskii ²

¹ School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
² Stavanger University, NO-4036 Stavanger, Norway

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     title = {Universal sampling of band-limited signals},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {927--931},
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     year = {2006},
     doi = {10.1016/j.crma.2006.04.015},
     language = {en},
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Alexander Olevskii; Alexander Ulanovskii. Universal sampling of band-limited signals. Comptes Rendus. Mathématique, Volume 342 (2006) no. 12, pp. 927-931. doi : 10.1016/j.crma.2006.04.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.04.015/

Bibliographie
Cité par

[1] A. Beurling Balayage of Fourier–Stietjes transforms, Collected Works of Arne Beurling, vol. 2, Harmonic Analysis, Birkhäuser, Boston, 1989

[2] A. Beurling; P. Malliavin On the closure of characters and the zeros of entire functions, Acta Math., Volume 118 (1967), pp. 79-93

[3] L. Bezuglaya; V. Katsnelson The sampling theorem for functions with limited multi-band spectrum, Z. Anal. Anwendungen, Volume 12 (1993) no. 3, pp. 511-534

[4] J. Bourgain; L. Tzafriri Invertibility of “large” submatrices with applications to the geometry of Banach spaces and harmonic analysis, Israel J. Math., Volume 57 (1987) no. 2, pp. 137-224

[5] J.R. Higgins Sampling Theory in Fourier and Signal Analysis. Foundations, Clarendon Press, Oxford, 1996

[6] H.J. Landau A sparse regular sequence of exponentials closed on large sets, Bull. Amer. Math. Soc., Volume 70 (1964), pp. 566-569

[7] H.J. Landau Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., Volume 117 (1967), pp. 37-52

[8] Yu. Lyubarskii; I. Spitkovsky Sampling and interpolation for a lacunary spectrum, Proc. Roy. Soc. Edinburgh Sect. A, Volume 126 (1996) no. 1, pp. 77-87

[9] A. Olevskii; A. Ulanovskii Almost integer translates. Do nice generators exist?, J. Fourier Anal. Appl., Volume 10 (2004) no. 1, pp. 93-104

[10] K. Seip Interpolation and Sampling in Spaces of Analytic Functions, University Lecture Series, vol. 33, American Mathematical Society, Providence, RI, 2004

[11] E. Szemerédi On sets of integers containing no k elements in arithmetic progression, Acta Arith., Volume 27 (1975), pp. 199-245

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