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Qualitative uncertainty principle for continuous modulated shearlet transform

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Abstract

We prove the qualitative uncertainty principle for the continuous modulated shearlet transform on several classes of groups including Abelian groups, compact extensions of Abelian groups and Heisenberg group. As particular cases, one obtains the qualitative uncertainty principles for the Gabor transform, the wavelet transform and the shearlet transform.

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References

  1. Bansal, A., Kumar, A.: Qualitative uncertainty principle for Gabor transform. Bull. Korean Math. Soc. 54(1), 71–84 (2017)

    Article  MathSciNet  Google Scholar 

  2. Bansal, P., Kumar, A., Bansal, A.: Continuous modulated Shearlet transform. Adv. Pure Appl. Math. 13(4), 29–57 (2022)

    Article  MathSciNet  Google Scholar 

  3. Benedek, A., Panzone, R.: The space \(L^p\), with mixed norm. Duke Math. J. 28(3), 301–324 (1961)

    Article  MathSciNet  Google Scholar 

  4. Cowling, M., Price, J.F., Sitaram, A.: A qualitative uncertainty principle for semisimple Lie groups. J. Aust. Math. Soc. 45(1), 127–132 (1988)

    Article  MathSciNet  Google Scholar 

  5. Echterhoff, S., Kaniuth, E., Kumar, A.: A qualitative uncertainty principle for certain locally compact groups. Forum Math. 3(3), 355–370 (1991)

    Article  MathSciNet  Google Scholar 

  6. Farashahi, A.G.: Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups. Banach J. Math. Anal. 11(1), 50–71 (2017)

    Article  MathSciNet  Google Scholar 

  7. Farashahi, A.G., Kamyabi-Gol, R.: Continuous Gabor transform for a class of non-Abelian groups. Bull. Belg. Math. Soc. Simon Stevin 19, 683–701 (2012)

    Article  MathSciNet  Google Scholar 

  8. Hogan, J.A.: A qualitative uncertainty principle for unimodular groups of type I. Trans. Am. Math. Soc. 340(2), 587–594 (1993)

    MathSciNet  Google Scholar 

  9. Kleppner, A., Lipsman, R.L.: The Plancherel formula for group extension. Ann. Sci. Ec. Norm. Super. 5, 459–516 (1972)

    Article  MathSciNet  Google Scholar 

  10. Mejjaoli, H., Hamadi, N.B., Omri, S.: Localization operators, time frequency concentration and quantitative-type uncertainty for the continuous wavelet transform associated with spherical mean operator. Int. J. Wavelets Multiresolut. Inf. Process. 17(4), 1–36 (2019)

    Article  MathSciNet  Google Scholar 

  11. Moore, C.C.: Groups with finite dimensional irreducible representation. Trans. Am. Math. Soc. 166, 401–410 (1972)

    Article  MathSciNet  Google Scholar 

  12. Nefzi, B., Brahim, K., Fitouhi, A.: Uncertainty principles for the multivariate continuous shearlet transform. J. Pseudo-Differ. Oper. Appl. 11, 517–542 (2020)

    Article  MathSciNet  Google Scholar 

  13. Price, J.F., Sitaram, A.: Functions and their Fourier transforms with supports of finite measure for certain locally compact groups. J. Funct. Anal. 79, 166–182 (1988)

    Article  MathSciNet  Google Scholar 

  14. Royden, H.L., Fitzpatrick, P.M.: Real Analysis, 4th edn. PHI Learning Pvt. Ltd. (2012)

  15. Sharma, J., Kumar, A.: Qualitative uncertainty principle for the Gabor transform on certain locally compact groups. Adv. Pure Appl. Math. 9(3), 205–220 (2018)

    Article  MathSciNet  Google Scholar 

  16. Smaoui, K., Abid, K.: Hardy’s theorem for Gabor transform on nilpotent Lie groups. J. Fourier Anal. Appl. 28(3), Paper No. 56 (2022)

    Article  MathSciNet  Google Scholar 

  17. Wilczok, E.: New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform. Doc. Math. 5, 201–226 (2000)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The second author acknowledges support from National Academy of Sciences, India.

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Authors and Affiliations

  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Piyush Bansal & Ajay Kumar

  2. Department of Mathematics, Keshav Mahavidyalaya (University of Delhi), H-4-5 Zone, Pitampura, Delhi, 110034, India

    Ashish Bansal

Authors
  1. Piyush Bansal
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  2. Ajay Kumar
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  3. Ashish Bansal
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Corresponding author

Correspondence to Ashish Bansal.

Additional information

Communicated by Chi-Keung Ng.

Dedicated to Ajit Iqbal Singh on her 80th birthday.

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Bansal, P., Kumar, A. & Bansal, A. Qualitative uncertainty principle for continuous modulated shearlet transform. Adv. Oper. Theory 9, 46 (2024). https://doi.org/10.1007/s43036-024-00346-5

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  • DOI: https://doi.org/10.1007/s43036-024-00346-5

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