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Journal of Fourier Analysis and Applications

, Volume 22, Issue 4, pp 768–786 | Cite as

An Uncertainty Principle and Sampling Inequalities in Besov Spaces

  • Philippe Jaming
  • Eugenia Malinnikova
Article

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.

Keywords

Uncertainty principle Sampling theory Besov spaces 

Notes

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.

References

  1. 1.
    Aldroubi, A.: Non-uniform weighted average sampling and reconstruction in shift-invariant and wavelet spaces. Appl. Comput. Harmon. Anal. 13, 151–161 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aldroubi, A., Gröchenig, K.: Nonuniform sampling and reconstruction in shift-invariant spaces. SIAM Rev. 43, 585–620 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourdaud, G.: Ondelettes et espaces de Besov. Rev. Mat. Iberoam. 11, 477–512 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cowling, M.G., Price, J.F.: Bandwidth versus time concentration: the Heisenberg-Pauli-Weyl inequality. SIAM J. Math. Anal. 15, 151–165 (1984)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Daubechies, I. : Ten lectures on wavelets. In: CBMS-NSF regional conference series in applied mathematics, vol. 61, SIAM (1992)Google Scholar
  6. 6.
    Feichtinger, H., Gröchenig, K.: Iterative reconstruction of multivariate band-limited functions from irregular sampling values. SIAM J. Math. Anal. 23(1), 244–261 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Feichtinger, H., Gröchenig, K.: Irregular sampling theorems and series expansion of band-limited functions. J. Math. Anal. Appl. 167, 530–556 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gröchenig, K.: Reconstruction algorithms in irregular sampling. Math. Comput. 59, 181–194 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gröchenig, K.: Irregular sampling of wavelet and short-time Fourier transforms. Constr. Approx. 9, 283–297 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gröchenig, K., Romero, J.L., Unnikrishnan, J., Vetterli, M.: On minimal trajectories for mobile sampling of bandlimited fields. Appl. Comput. Harmon. Anal. doi: 10.1016/j.acha.2014.11.002
  11. 11.
    Jaffard, S., Meyer, Y.: On the regularity of functions in critical Besov spaces. J. Funct. Anal. 175, 415–434 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jaffard, S., Okada, M., Ueno, T.: Approximate sampling theorem and the order of smoothness of the Besov space, harmonic analysis and nonlinear partial differential equations, 45–56, RIMS Kôkyûroku Bessatsu, B18, Research Institute for Mathematics and Science (RIMS), Kyoto (2010)Google Scholar
  13. 13.
    Kyriazis, G.: Wavelet coefficients measuring smoothness in \(H^p({\mathbb{R}}^d)\). Appl. Comput. Harmon. Anal. 3, 100–119 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lyubarskii, Yu., Madych, W.R.: The recovery of irregular sampled band limited functions via tempered splines. J. Funct. Anal. 125, 201–222 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Madych, W.R.: An estimate for multivariate interpolation, II. J. Approx. Theory 142, 116–128 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Martin, J., Milman, M.: Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces. arXiv:1501.06556 [math.FA]
  17. 17.
    Meyer, Y.: Ondelettes et opérateurs. Hermann, Paris (1990)zbMATHGoogle Scholar
  18. 18.
    Peetre, J.: New Thoughts on Besov Spaces. Mathematics Series. Duke University, Durham (1976)zbMATHGoogle Scholar
  19. 19.
    Pesenson, I.: A reconstruction formula for band limited functions in \(L^2(\mathbb{R}^d)\). Proc. AMS 127, 3593–3600 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Pesenson, I.: Plancherel-Polya-type inequalities for entire functions of exponential type in \(L_p(\mathbb{R}^d)\). J. Math. Anal. Appl. 330, 1194–1206 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Strichartz, R.S.: Uncertainty principles in harmonic analysis. J. Funct. Anal. 84, 97–114 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Triebel, H.: Characterizations of Besov-Hardy-Sobolev spaces via harmonic functions, temperatures, and related means. J. Approx. Theory 35, 275–297 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Triebel, H.: Theory of Function Spaces III. Monographs in Mathematics. Birkhaüser, Basel (2006)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Univ. Bordeaux, IMB, UMR 5251TalenceFrance
  2. 2.CNRS, IMB, UMR 5251TalenceFrance
  3. 3.Department of Mathematical SciencesNorwegian University of Science and Technology (NTNU)TrondheimNorway

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