Necessary conditions for Schatten class localization operators
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- by Elena Cordero and Karlheinz Gröchenig PDF
- Proc. Amer. Math. Soc. 133 (2005), 3573-3579 Request permission
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
- F. A. Berezin, Wick and anti-Wick symbols of operators, Mat. Sb. (N.S.) 86(128) (1971), 578–610 (Russian). MR 0291839
- Paolo Boggiatto and Elena Cordero, Anti-Wick quantization of tempered distributions, Progress in analysis, Vol. I, II (Berlin, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 655–662. MR 2032736
- Elena Cordero and Karlheinz Gröchenig, Time-frequency analysis of localization operators, J. Funct. Anal. 205 (2003), no. 1, 107–131. MR 2020210, DOI 10.1016/S0022-1236(03)00166-6
- Ingrid Daubechies, Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), no. 4, 605–612. MR 966733, DOI 10.1109/18.9761
- F. De Mari, H. G. Feichtinger, and K. Nowak, Uniform eigenvalue estimates for time-frequency localization operators, J. London Math. Soc. (2) 65 (2002), no. 3, 720–732. MR 1895743, DOI 10.1112/S0024610702003101
- H. G. Feichtinger. Modulation spaces on locally compact abelian groups. In Proceedings of “International Conference on Wavelets and Applications" 2002, Chennai, India. Updated version of a technical report, University of Vienna, 1983.
- Hans G. Feichtinger and K. Gröchenig, Gabor frames and time-frequency analysis of distributions, J. Funct. Anal. 146 (1997), no. 2, 464–495. MR 1452000, DOI 10.1006/jfan.1996.3078
- Hans G. Feichtinger and Thomas Strohmer (eds.), Advances in Gabor analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1955929, DOI 10.1007/978-1-4612-0133-5
- Gerald B. Folland, Harmonic analysis in phase space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ, 1989. MR 983366, DOI 10.1515/9781400882427
- Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
- Roger Howe, Quantum mechanics and partial differential equations, J. Functional Analysis 38 (1980), no. 2, 188–254. MR 587908, DOI 10.1016/0022-1236(80)90064-6
- Jayakumar Ramanathan and Pankaj Topiwala, Time-frequency localization via the Weyl correspondence, SIAM J. Math. Anal. 24 (1993), no. 5, 1378–1393. MR 1234023, DOI 10.1137/0524080
- Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959
- Richard Rochberg and Stephen Semmes, Nearly weakly orthonormal sequences, singular value estimates, and Calderon-Zygmund operators, J. Funct. Anal. 86 (1989), no. 2, 237–306. MR 1021138, DOI 10.1016/0022-1236(89)90054-2
- Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
- Kristian Seip, Density theorems for sampling and interpolation in the Bargmann-Fock space. I, J. Reine Angew. Math. 429 (1992), 91–106. MR 1173117, DOI 10.1515/crll.1992.429.91
- M. A. Shubin, Pseudodifferential operators and spectral theory, 2nd ed., Springer-Verlag, Berlin, 2001. Translated from the 1978 Russian original by Stig I. Andersson. MR 1852334, DOI 10.1007/978-3-642-56579-3
- Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149, DOI 10.1007/BFb0064579
- Joachim Toft, Continuity properties for modulation spaces, with applications to pseudo-differential calculus. I, J. Funct. Anal. 207 (2004), no. 2, 399–429. MR 2032995, DOI 10.1016/j.jfa.2003.10.003
- M. W. Wong, Wavelet transforms and localization operators, Operator Theory: Advances and Applications, vol. 136, Birkhäuser Verlag, Basel, 2002. MR 1918652, DOI 10.1007/978-3-0348-8217-0
Additional Information
- Elena Cordero
- Affiliation: Department of Mathematics, Politecnico of Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
- MR Author ID: 629702
- 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
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 133 (2005), 3573-3579
- MSC (2000): Primary 35S05, 47G30, 46E35, 47B10
- DOI: https://doi.org/10.1090/S0002-9939-05-07897-4
- MathSciNet review: 2163592