Revisiting Landauʼs density theorems for Paley–Wiener spaces

Shahaf Nitzan; Alexander Olevskii

doi:10.1016/j.crma.2012.05.003

Harmonic Analysis

Revisiting Landauʼs density theorems for Paley–Wiener spaces
[Retour sur les théorèmes de densité de Landau dans les espaces de Paley–Wiener]

Présenté par : Jean-Pierre Kahane

Shahaf Nitzan ¹ ; Alexander Olevskii ²

¹ School of Mathematical Sciences, Weizmann Institute for Science, Rehovot 76100, Israel
² School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 509-512.

Résumé
Abstract

On présente ici une approche simple, et plutôt surprenante, des théorèmes de densité de Landau fournissant, pour lʼéchantillonnage et lʼinterpolation, des versions plus fortes des résultats connus. En particulier, on étend le théorème dʼinterpolation au spectre non borné.

We present a surprisingly simple approach to Landauʼs density theorems for sampling and interpolation, which provides stronger versions of these results. In particular, we extend the interpolation theorem to unbounded spectra.

Reçu le : 2012-04-03
Accepté le : 2012-05-11
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.05.003

Affiliations des auteurs :

Shahaf Nitzan ¹ ; Alexander Olevskii ²

¹ School of Mathematical Sciences, Weizmann Institute for Science, Rehovot 76100, Israel
² School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv 69978, Israel

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     title = {Revisiting {Landau's} density theorems for {Paley{\textendash}Wiener} spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {509--512},
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     year = {2012},
     doi = {10.1016/j.crma.2012.05.003},
     language = {en},
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Shahaf Nitzan; Alexander Olevskii. Revisiting Landauʼs density theorems for Paley–Wiener spaces. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 509-512. doi : 10.1016/j.crma.2012.05.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.05.003/

Bibliographie
Cité par

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[2] K. Gröchenig; H. Razafinjatovo On Landauʼs necessary density conditions for sampling and interpolation of band limited functions, J. Lond. Math. Soc., Volume 54 (1996) no. 3, pp. 557-565

[3] J.-P. Kahane Sur les fonctions moyenne-periodiques bornées, Ann. Inst. Fourier, Volume 7 (1957), pp. 293-314

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

[5] A. Olevskii; A. Ulanovskii Interpolation in Bernstein and Paley–Wiener spaces, J. Funct. Anal., Volume 256 (2009) no. 10, pp. 3257-3278

[6] A. Olevskii; A. Ulanovskii Approximation of discrete functions and size of spectrum, St. Petersburg Math. J., Volume 21 (2010), pp. 1015-1025

[7] J. Ramanathan; T. Steger Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., Volume 2 (1995) no. 2, pp. 148-153

[8] R.M. Young An Introduction to Nonharmonic Fourier Series, Academic Press, 2001

Cité par Sources :

^☆ Research supported in part by Israel Science Foundation.

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