Harmonic Analysis/Mathematical Analysis
Interpolation by functions with small spectra

Presented by: Jean-Pierre Kahane

Olevskii, Alexander1; Ulanovskii, Alexander2
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.

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 L2-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:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.07.005
Olevskii, Alexander 1; Ulanovskii, Alexander 2

1 School of Mathematics, Tel Aviv University, Ramat Aviv, 69978 Israel
2 Stavanger University, N-4036 Stavanger, Norway 
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Olevskii, Alexander; Ulanovskii, Alexander. 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. http://archive.numdam.org/articles/10.1016/j.crma.2007.07.005/

[1] Beurling, A. Interpolation for an interval in R1, The Collected Works of Arne Beurling, Harmonic Analysis, vol. 2, Birkhäuser, Boston, 1989

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

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

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