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The Donoho–Stark, Benedicks and Heisenberg type uncertainty principles, and the localization operators for the Heckman–Opdam continuous wavelet transform on \({\mathbb {R}}^{d}\)

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Abstract

We consider the continuous wavelet transform \(\Phi _{h}^{W}\) associated with the Heckman–Opdam operators on \(\mathbb {R}^{d}\). We analyse the concentration of this transform on sets of finite measure. In particular, Donoho–Stark and Benedicks-type uncertainty principles are given. Next, we prove many versions of Heisenberg-type uncertainty principles for \(\Phi _{h}^{W}\). Finally, we investigate the localization operators for \(\Phi _{h}^{W}\), in particular we prove that they are in the Schatten–von Neumann class.

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Acknowledgements

The authors are deeply indebted to the referees for providing constructive comments and helps in improving the contents of this article.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, Taibah University, PO BOX 30002, Al-Madina Al-Munawarah, Saudi Arabia

    Hatem Mejjaoli

  2. Department of Mathematics, Faculty of Sciences of Tunis, University of El Manar, Campus, 2092, Tunis, Tunisia

    Khalifa Trimèche

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  1. Hatem Mejjaoli
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  2. Khalifa Trimèche
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Correspondence to Hatem Mejjaoli.

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Mejjaoli, H., Trimèche, K. The Donoho–Stark, Benedicks and Heisenberg type uncertainty principles, and the localization operators for the Heckman–Opdam continuous wavelet transform on \({\mathbb {R}}^{d}\) . J. Pseudo-Differ. Oper. Appl. 8, 423–452 (2017). https://doi.org/10.1007/s11868-017-0194-z

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  • DOI: https://doi.org/10.1007/s11868-017-0194-z

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