Abstract
In this paper, we study the conjecture of Bansal, Kumar and Sharma, which is an analog of Hardy’s theorem for Gabor transform in the setup of connected nilpotent Lie groups. To approach this conjecture, we use the orbit method and the Plancherel theory. When the Lie group G is simply connected, we show that the conjecture is true.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Astengo, F., Cowling, M., di Blasio, B., Sundari, M.: Hardy’s uncertainty principle on certain Lie groups. J. Lond. Math. Soc. 62, 461–472 (2000)
Baklouti, A., Kaniuth, E.: On Hardy’s uncertainty principle for connected nilpotent Lie groups. Math. Z. 259, 233–247 (2008)
Baklouti, A., Kaniuth, E.: On Hardy’s uncertainty principle for solvable locally compact groups. J. Fourier Anal. Appl. 16, 129–147 (2010)
Baklouti, A., Smaoui, K., Ludwig, J.: Estimate of \(L^p\)-Fourier transform norm on nilpotent Lie groups. J. Funct. Anal. 199, 508–520 (2003)
Bansal, A., Kumar, A.: Heisenberg uncertainty inequality for Gabor transform. J. Math. Inequal. 10, 737–749 (2016)
Bansal, A., Kumar, A., Sharma, J.: Hardy’s theorem for Gabor transform. J. Aust. Math. Soc. 106, 143–159 (2018)
Bonami, A., Demange, B., Jaming, P.: Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19, 23–55 (2003)
Corwin, L., Greenleaf, F.P.: Representations of Nilpotent Lie Groups and their Applications, Part 1: Basic Theory and Examples. Cambridge University Press, Cambridge (1990)
Cowling, M., Sitaram, A., Sundari, M.: Hardy’s uncertainty principle on semisimple Lie groups. Pac. J. Math. 192, 293–296 (2000)
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)
Gröchenig, K.: Foundation of Time-Frequency Analysis. Birkhäser, Boston (2000)
Gröchenig, K.: Uncertainty principles for time–frequency representations. In: Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2003)
Hardy, G.H.: A theorem concerning Fourier transforms. J. Lond. Math. Soc. 8, 227–231 (1933)
Kaniuth, E., Kumar, A.: Hardy’s theorem for simply connected nilpotent Lie groups. Math. Proc. Camb. Philos. Soc. 131, 487–494 (2001)
Sarkar, R.P., Thangavelu, S.: A complete analogue of Hardy’s theorem on semisimple Lie groups. Colloq. Math. 93, 27–40 (2002)
Sarkar, R.P., Thangavelu, S.: On theorems of Beurling and Hardy for the Euclidean motion group. Tohoku Math. J. 57, 335–351 (2005)
Sengupta, J.: An analogue of Hardy’s theorem for semi-simple Lie groups. Proc. Am. Math. Soc. 128, 2493–2499 (2000)
Sitaram, A., Sundari, M.: An analogue of Hardy’s theorem for very rapidly decreasing functions on semisimple groups. Pac. J. Math. 177, 187–200 (1997)
Sitaram, A., Sundari, M., Thangavelu, S.: Uncertainty principles on certain Lie groups. Proc. Indiana Acad. Sci. Math. Sci. 105, 135–151 (1995)
Sundari, M.: Hardy’s theorem for the n-dimensional Euclidean motion group. Proc. Am. Math. Soc. 126, 1199–1204 (1998)
Thangavelu, S.: An analogue of Hardy’s theorem for the Heisenberg group. Colloq. Math. 87, 137–145 (2001)
Thangavelu, S.: An Introduction to the Uncertainty Principle. Hardy’s Theorem on Lie Groups, Birkhäuser, Boston (2003)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Sundaram Thangavelu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Smaoui, K., Abid, K. Hardy’s Theorem for Gabor Transform on Nilpotent Lie Groups. J Fourier Anal Appl 28, 56 (2022). https://doi.org/10.1007/s00041-022-09949-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-022-09949-z