Quasi-frames of translates

Nitzan, Shahaf; Olevskii, Alexander

doi:10.1016/j.crma.2009.04.001

Mathematical Analysis/Harmonic Analysis

Quasi-frames of translates

Presented by: Jean-Pierre Kahane

Nitzan, Shahaf ¹; Olevskii, Alexander ¹

¹ School of Mathematical Sciences, Tel Aviv University, Ramat-Aviv, Israel 69978

Comptes Rendus. Mathématique, Volume 347 (2009) no. 13-14, pp. 739-742.

Abstract
Résumé

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 $f \in L^{2} (R)$ can be approximated with arbitrary small $L^{2}$ -error by a linear combination $\sum c_{λ} g (t - λ)$ satisfying the $l_{q}$ -estimate of the coefficients:

‖ {c_{λ}} ‖_{l_{q}} ⩽ 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 $g \in L^{2} (R)$ , telles que le système des translatées ${g (t - λ)} (λ \in Λ)$ soit un “quasi-frame” au sens suivant : pour tout $q > 2$ il existe $C (q) > 0$ tel que toute fonction $f \in L^{2} (R)$ est approchable dans $L^{2} (R)$ par des combinaisons linéaires $\sum c_{λ} g (t - λ)$ vérifiant ${(\sum {| c_{λ} |}^{q})}^{1 / q} ⩽ C (q) ‖ f ‖$ . Cela est impossible quand $q = 2$ , selon un résultat de Christensen, Deng et Heil.

Received: 2009-03-12
Accepted: 2009-03-31
Published online: 2009-05-28

DOI: 10.1016/j.crma.2009.04.001

Author's affiliations:

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, Volume 347 (2009) no. 13-14, pp. 739-742. doi : 10.1016/j.crma.2009.04.001. http://www.numdam.org/articles/10.1016/j.crma.2009.04.001/

References
Cited by

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[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 $L^{2} (R)$ of almost integer translates, C. R. Acad. Sci. Paris, Ser. I, Volume 324 (1997), pp. 987-991

[7] Olevskii, A. Approximation by translates in $L^{2} (R)$ , 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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^☆ Supported in part by the Israel Science Foundation.