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Necessary conditions for Schatten class localization operators

Authors: Elena Cordero and Karlheinz Gröchenig
Journal: Proc. Amer. Math. Soc. 133 (2005), 3573-3579
MSC (2000): Primary 35S05, 47G30, 46E35, 47B10
Published electronically: June 6, 2005
MathSciNet review: 2163592
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Abstract: We study time-frequency localization operators of the form $A_a^{\varphi_1,\varphi_2}$, where $a$ is the symbol of the operator and $\varphi_1 , \varphi_2 $ are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for $A_a^{\varphi_1,\varphi_2}\in S_p(L^2(\mathbb{R} ^d))$, the Schatten class of order $p$, is that $a$ belongs to the modulation space $M^{p,\infty}(\mathbb{R} ^{2d})$ and the window functions to the modulation space $M^1$. Here we prove a partial converse: if $A_a^{\varphi_1,\varphi_2}\in S_p(L^2(\mathbb{R} ^d))$ for every pair of window functions $\varphi_1,\varphi_2\in \mathcal{S}(\mathbb{R} ^{2d})$ with a uniform norm estimate, then the corresponding symbol $a$ must belong to the modulation space $M^{p,\infty}(\mathbb{R} ^{2d})$. In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For $p=\infty$ and $p=2$, we recapture earlier results, which were obtained by different methods.

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Additional Information

Elena Cordero
Affiliation: Department of Mathematics, Politecnico of Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Karlheinz Gröchenig
Affiliation: Institute of Biomathematics and Biometry, GSF - National Research Center for Environment and Health, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany

Keywords: Localization operator, modulation space, short-time Fourier transform, Schatten class operator
Received by editor(s): April 7, 2004
Received by editor(s) in revised form: July 12, 2004
Published electronically: June 6, 2005
Additional Notes: The first author was supported in part by the national project MIUR “Analisi Armonica” (coordination by G. Mauceri).
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society

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