Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 35S05, 47G30, 46E35, 47B10

Retrieve articles in all journals with MSC (2000): 35S05, 47G30, 46E35, 47B10


Additional Information

Elena Cordero
Affiliation: Department of Mathematics, Politecnico of Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
Email: cordero@dm.unito.it

Karlheinz Gröchenig
Affiliation: Institute of Biomathematics and Biometry, GSF - National Research Center for Environment and Health, Ingolstädter Landstraße 1, 85764 Neuherberg, Germany
Email: karlheinz.groechenig@gsf.de

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07897-4
PII: S 0002-9939(05)07897-4
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