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An Uncertainty Principle and Sampling Inequalities in Besov Spaces

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Abstract

We extend Strichartz’s uncertainty principle (Strichartz, J Funct Anal 84:97–114, 1989) from the setting of the Sobolev space \(W^{1,2}({\mathbb {R}})\) to more general Besov spaces \(B^{1/p}_{p,1}({\mathbb {R}})\). The main result gives an estimate from below of the trace of a function from the Besov space on a uniformly distributed discrete subset. We also prove the corresponding result in the multivariate case and discuss some applications to irregular approximate sampling in critical Besov spaces.

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Acknowledgments

This work was mostly executed while the first author visited the Norwegian University of Science and Technology and the second author visited University of Bordeaux. It is our pleasure to thank both Departments for hospitality. We are also grateful to an anonymous referee for useful comments and suggestions. Ph.J. kindly acknowledge financial support from the French ANR programs ANR 2011 BS01 007 01 (GeMeCod), ANR-12-BS01-0001 (Aventures). This study has been carried out with financial support from the French State, managed by the French National Research Agency (ANR) in the frame of the Investments for the future Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02). E.M. was partly supported by Project 213638 of the Research Council of Norway. This research was sponsored by the French-Norwegian PHC AURORA 2014 PROJECT N 31887TC, N 233838, CHARGE.

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Authors and Affiliations

  1. Univ. Bordeaux, IMB, UMR 5251, 33400, Talence, France

    Philippe Jaming

  2. CNRS, IMB, UMR 5251, 33400, Talence, France

    Philippe Jaming

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), 7491, Trondheim, Norway

    Eugenia Malinnikova

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  1. Philippe Jaming
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  2. Eugenia Malinnikova
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Correspondence to Philippe Jaming.

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Communicated by Karlheinz Gröchenig.

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Jaming, P., Malinnikova, E. An Uncertainty Principle and Sampling Inequalities in Besov Spaces. J Fourier Anal Appl 22, 768–786 (2016). https://doi.org/10.1007/s00041-015-9435-9

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  • DOI: https://doi.org/10.1007/s00041-015-9435-9

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