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 -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 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).
Accepted:
Published online:
Alexander Olevskii 1; Alexander Ulanovskii 2
@article{CRMATH_2007__345_5_261_0, author = {Alexander Olevskii and Alexander Ulanovskii}, title = {Interpolation by functions with small spectra}, journal = {Comptes Rendus. Math\'ematique}, pages = {261--264}, publisher = {Elsevier}, volume = {345}, number = {5}, year = {2007}, doi = {10.1016/j.crma.2007.07.005}, language = {en}, }
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.005/
[1] Interpolation for an interval in , The Collected Works of Arne Beurling, Harmonic Analysis, vol. 2, Birkhäuser, Boston, 1989
[2] Sur les fonctions moyenne-périodiques bornées, Ann. Inst. Fourier, Volume 7 (1957), pp. 293-314
[3] Necessary density conditions for sampling and interpolation of certain entire functions, Acta Math., Volume 117 (1967), pp. 37-52
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