Interpolation by functions with small spectra

Alexander Olevskii; Alexander Ulanovskii

doi:10.1016/j.crma.2007.07.005

Harmonic Analysis/Mathematical Analysis

Interpolation by functions with small spectra

Presented by: Jean-Pierre Kahane

Alexander Olevskii ¹; Alexander Ulanovskii ²

¹ School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Israel
² Stavanger University, N-4036 Stavanger, Norway

Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 261-264

Abstract (Original language)
Résumé

We show that if Λ is a ‘generic’ separated sequence of reals, then there is an unbounded set S of arbitrary small measure (union of some neighborhoods of integers) such that every function on Λ with certain decay condition, can be interpolated by an $L^{2}$ -function with the spectrum on S (Theorem 1). This should be contrasted against results for compact spectra (Theorems 2 and 3).

Nous montrons que si Λ est une suite réelle « générique », il existe un ensemble S de mesure arbitrairement petite et non borné (réunion de voisinages d'entiers) tel que toute fonction à décroissance convenable sur Λ soit prolongeable sur $R$ en une fonction de carré integrable dont le spectre est dans S (Théorème 1). Cela doit être comparé aux résultats concernant les spectres compacts (Théorèmes 2 et 3).

Received: 2007-06-29
Accepted: 2007-07-18
Published online: 2007-08-21

DOI: 10.1016/j.crma.2007.07.005

Author's affiliations:

Alexander Olevskii ¹; Alexander Ulanovskii ²

¹ School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Israel
² Stavanger University, N-4036 Stavanger, Norway

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     title = {Interpolation by functions with small spectra},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {261--264},
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     number = {5},
     doi = {10.1016/j.crma.2007.07.005},
     language = {en},
}

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Alexander Olevskii; Alexander Ulanovskii. Interpolation by functions with small spectra. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 261-264. doi: 10.1016/j.crma.2007.07.005

References
Cited by

[1] A. Beurling Interpolation for an interval in $R^{1}$ , The Collected Works of Arne Beurling, Harmonic Analysis, vol. 2, Birkhäuser, Boston, 1989

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

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

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