no. 13-14
Mathematical Analysis/Harmonic Analysis
Quasi-frames of translates
[Systèmes de translatées proches des frames]
Présenté par : Jean-Pierre Kahane
Nitzan, Shahaf1 ; Olevskii, Alexander1
1 School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv, Israel 69978
Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 739-742

We construct uniformly discrete, and even sparse, sequences of translates {g(t−λ)} of a single function which have the following frame-type approximation property: for every q>2 there exists C(q) such that every function f∈L2(R) can be approximated with arbitrary small L2-error by a linear combination ∑cλg(t−λ) satisfying the lq-estimate of the coefficients:

‖{cλ}‖lq⩽C(q)‖f‖.
This cannot be done for q=2, according to a result of Christensen, Deng and Heil.

Nous construisons une suite réelle Λ uniformément discrète (de pas >0) et même lacunaire, et une fonction g∈L2(R), telles que le système des translatées {g(t−λ)}(λ∈Λ) soit un “quasi-frame” au sens suivant : pour tout q>2 il existe C(q)>0 tel que toute fonction f∈L2(R) est approchable dans L2(R) par des combinaisons linéaires ∑cλg(t−λ) vérifiant (∑|cλ|q)1/q⩽C(q)‖f‖. Cela est impossible quand q=2, selon un résultat de Christensen, Deng et Heil.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.04.001

Nitzan, Shahaf 1 ; Olevskii, Alexander 1

1 School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv, Israel 69978 
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Nitzan, Shahaf; Olevskii, Alexander. Quasi-frames of translates. Comptes Rendus. Mathématique, Tome 347 (2009) no. 13-14, pp. 739-742. doi: 10.1016/j.crma.2009.04.001

[1] Christensen, O.; Deng, B.; Heil, C. Density of Gabor frames, Appl. Comput. Harmon. Anal., Volume 7 (1999), pp. 292-304

[2] Katznelson, Y. An Introduction to Harmonic Analysis, Dover Publications, Inc., New York, 1976

[3] Kozma, G.; Olevskii, A. Menshov representation spectra, J. Anal. Math., Volume 84 (2001), pp. 361-393

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

[5] Nitzan-Hahamov, S.; Olevskii, A. Sparse exponential systems: completeness with estimates, Israel J. Math., Volume 158 (2007), pp. 205-215

[6] Olevskii, A. Completeness in L2(R) of almost integer translates, C. R. Acad. Sci. Paris, Ser. I, Volume 324 (1997), pp. 987-991

[7] Olevskii, A. Approximation by translates in L2(R), Real Anal. Exchange, Volume 24 (1998/1999) no. 1, pp. 43-44

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

[9] Ramanathan, J.; Steger, T. Incompleteness of sparse coherent states, Appl. Comput. Harmon. Anal., Volume 2 (1995), pp. 148-153

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