Advertisement

Uncertainty principles for the multivariate continuous shearlet transform

  • Bochra NefziEmail author
  • Kamel Brahim
  • Ahmed Fitouhi
Article
  • 9 Downloads

Abstract

In this paper, we present some new elements of harmonic analysis related to multivariate continuous shearlet transform introduced earlier in Dahlke et al. (J Fourier Anal Appl 16:340–364, 2010; The continuous shearlet transform in arbitrary space dimensions, Philipps-Universität Marburg, Marburg, 2008). Thus, some results (Parseval’s formula, inversion formula, etc.) are established. Next, we prove an analogue of Heisenberg’s inequality for shearlet transform. Last, we study shearlet transform on subset of finite measures.

Keywords

Fourier transform Shearlet The multivariate continuous shearlet transform Uncertainty principle Heisenberg uncertainty inequality Local uncertainty inequality 

Mathematics Subject Classification

33B15 33D05 44A20 42A38 42B10 

Notes

References

  1. 1.
    Amrein, W.O., Berthier, A.M.: On support properties of \(L^p\)-functions and their Fourier transforms. J. Funct. Anal. 24, 258–267 (1977)CrossRefzbMATHGoogle Scholar
  2. 2.
    Benedicks, M.: On Fourier transforms of functions supported on sets of finite Lebesgue measure. J. Math. Anal. Appl. 106, 180–183 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoam. 19, 23–55 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bowie, P.C.: Uncertainty inequalities for Hankel transforms. SIAM J. Math. Anal. 2, 601–606 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ciatti, P., Ricci, F., Sundari, M.: Heisenberg–Pauli–Weyl uncertainty inequalities and polynomial volume growth. Adv. Math. 215(2), 616–625 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    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
  7. 7.
    Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.-G., Teschke, G.: The uncertainty principle associated with the continuous shearlet transform. Int. J. Wavelets Multiresolut. Inf. Process. 6, 157–181 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dahlke, S., Steidl, G., Teschke, G.: The continuous shearlet transform in arbitrary space dimensions. J. Fourier Anal. Appl. 16, 340–364 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dahlke, S., Steidl, G., Teschke, G.: The Continuous Shearlet Transform in Arbitrary Space Dimensions. Preprint Nr. 2008-7. Philipps-Universit\(\ddot{a}\)t Marburg, Marburg (2008)Google Scholar
  10. 10.
    Daubechies, I.: Time–frequency localization operators: a geometric phase space approach. IEEE Trans. Inf. Theory 34, 605–612 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Donoho, D.L., Strak, P.B.: Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49(3), 906–931 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet coorbit spaces and associated Banach frames. Appl. Comput. Harmon. Anal. 27, 195–214 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Faris, W.G.: Inequalities and uncertainty inequalities. J. Math. Phys. 19, 461–466 (1978)CrossRefGoogle Scholar
  14. 14.
    Folland, G.B., Sitaram, A.: The uncertainty principle: a mathematical survey. J. Fourier Anal. Appl. 3(3), 207–238 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hardy, G.A.: Theorem concerning Fourier transforms. J. Lond. Math. Soc. 1, 227–231 (1933)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Havin, V., Jöricke, B.: The Uncertainty Principle in Harmonic Analysis. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  17. 17.
    Heisenberg, W.: Uber den anschaulichen Inhalt der quantentheo-retischen Kinematik und Mechanik. Z. Phys. 43, 172–198 (1927)CrossRefzbMATHGoogle Scholar
  18. 18.
    Hogan, J.A.: Time–Frequency and Time-Scale Methods: Adaptive Decompositions, Uncertainty Principles, and Sampling. Springer, Basel (2007)Google Scholar
  19. 19.
    Ghobber, S., Jaming, P.: Uncertainty principles for integral operators. Stud. Math. 220(3), 197–220 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ghobber, S., Omri, S.: Time–frequency concentration of the windowed Hankel transform. Integral Transf. Spec. Funct. 25, 481–496 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Guo, K., Kutyniok, G., Labate, D.: Sparse multidimensional representations using anisotropic dilation and shear operators. In: Wavelets and Splines, pp. 189–201. Nashboro Press, Nashville, TN (2005)Google Scholar
  22. 22.
    Guo, K., Labate, D.: Characterization and analysis of edges using the continuous shearlet transform. SIAM J. Imaging Sci. 2, 959–986 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Guo, K., Labate, D.: Characterization of piecewise-smooth surfaces using the 3D continuous shearlet transform. J. Fourier Anal. Appl. 18, 488–516 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kennard, E.H.: Zur Quantenmechanik einfacher Bewegungstypen. Z. Phys. 44, 326–352 (1927)CrossRefzbMATHGoogle Scholar
  25. 25.
    Koornwinder, T.H.: Wavelets: an elementary treatment of theory and applications. In: Series in Approximations and Decompositions, vol. 3. World Scientific, Singapore (1993)Google Scholar
  26. 26.
    Kutyniok, G., Labate, D.: Analysis and identification of multidimensional singularities using the continuous shearlets transform. In: Kutyniok, G., Labate, D. (eds.) Shearlet, pp. 69–103. Birkhäuser, Boston (2012)CrossRefGoogle Scholar
  27. 27.
    Kutyniok, G., Labate, D.: Introduction to shearlets. In: Kutyniok, G., Labate, D. (eds.) Shearlet, pp. 1–38. Birkhäuser, Boston (2012)CrossRefGoogle Scholar
  28. 28.
    Kutyniok, G., Labate, D.: Resolution of the wavefront set using continuous shearlets. Trans. Am. Math. Soc. 361, 2719–2754 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Laugesen, R.S.S., Weaver, N., Weiss, G.L., Wilson, E.N.: A characterization of the higher dimensional groups associated with continuous wavelets. J. Geom. Anal. 12(1), 89–102 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Li, Y., Chen, R., Liang, S.: A new image denoising method based on shearlet shrinkage and improved total variation. Intell. Sci. Intell. Data Eng. 7202, 382–388 (2012)CrossRefGoogle Scholar
  31. 31.
    Liu, S., Hu, S., Xiao, Y., An, L.: A Bayesian shearlet shrinkage for SAR image denoising via sparse representation. Multidimens. Syst. Signal Process. 25, 683–701 (2014)CrossRefGoogle Scholar
  32. 32.
    Price, J.F.: Inequalities and local uncertainty principles. J. Math. Phys. 24, 1711–1714 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Price, J.F.: Sharp local uncertainty inequalities. Stud. Math. 85, 37–45 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Price, J.F., Sitaram, A.: Local uncertainty inequalities for locally compact groups. Trans. Am. Math. Soc. 308, 105–114 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Rösler, M., Voit, M.: Uncertainty principle for Hankel transforms. Proc. Am. Math. Soc. 127, 183–194 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Rudin, W.: Analyse r\(\acute{e}\)elle et complexe. DUNOD, Paris (1998)Google Scholar
  37. 37.
    Singer, P.: Uncertainty inequalities for the continuous wavelet transform. IEEE Trans. Inf. Theory 45, 1039–1042 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Stein, E.M.: Interpolation of linear operators. Trans. Am. Math. Soc. 83, 482–492 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)zbMATHGoogle Scholar
  40. 40.
    Weyl, H.: Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig. (Revised English edition: Groups and quantum mechanics). Dover (1950)Google Scholar
  41. 41.
    Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math. 5, 201–226 (2000)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Sciences of TunisUniversity of Tunis El ManarTunisTunisia
  2. 2.Department of Mathematics, College of ScienceUniversity of BishaBishaSaudi Arabia

Personalised recommendations