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Gabor System based on the unitary dual of the Heisenberg group

  • Santi R. Das and Radha Ramakrishnan EMAIL logo
Published/Copyright: March 25, 2025
Forum Mathematicum
From the journal Forum Mathematicum Volume 37 Issue 6

Abstract

In this paper, the Gabor system based on the unitary dual of the Heisenberg group H n is introduced and a sufficient condition is obtained for the Gabor system to be a Bessel sequence for L 2 ⁢ ( R ∗ , B 2 ; d ⁢ κ ) using the Schrödinger representation of H n , where B 2 denotes the class of Hilbert–Schmidt operators on L 2 ⁢ ( R n ) and d ⁢ κ denotes the Haar measure on R ∗ . Further, a necessary and sufficient condition is provided for the Gabor system to be an orthonormal system, a Parseval frame sequence, a frame sequence and a Riesz sequence.

Keywords: Bessel sequence; frames; Gabor system; Heisenberg group; Hilbert–Schmidt operator; orthonormal system; Parseval frame; Riesz basis; unitary representation; Weyl transform
MSC 2020: 42C15; 43A30; 47A67

Acknowledgements

The authors profusely thank Prof. K. Parthasarathy, RIASM, for his helpful suggestions regarding the explicit calculations for the Fourier transform on R ∗ .

  1. Communicated by: Guozhen Lu

References

[1] S. Arati and R. Radha, Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indag. Math. (N. S.) 30 (2019), no. 1, 106–127. 10.1016/j.indag.2018.09.001Search in Google Scholar

[2] S. Arati and R. Radha, Orthonormality of wavelet system on the Heisenberg group, J. Math. Pures Appl. (9) 131 (2019), 171–192. 10.1016/j.matpur.2019.02.004Search in Google Scholar

[3] S. Arati and R. Radha, Wavelet system and Muckenhoupt A 2 condition on the Heisenberg group, Colloq. Math. 158 (2019), no. 1, 59–76. 10.4064/cm7467-9-2018Search in Google Scholar

[4] D. Barbieri, E. Hernández and V. Paternostro, Spaces invariant under unitary representations of discrete groups, J. Math. Anal. Appl. 492 (2020), no. 1, Article ID 124357. 10.1016/j.jmaa.2020.124357Search in Google Scholar

[5] M. Bownik and K. A. Ross, The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl. 21 (2015), no. 4, 849–884. 10.1007/s00041-015-9390-5Search in Google Scholar

[6] C. Cabrelli and V. Paternostro, Shift-invariant spaces on LCA groups, J. Funct. Anal. 258 (2010), no. 6, 2034–2059. 10.1016/j.jfa.2009.11.013Search in Google Scholar

[7] O. Christensen, An Introduction to Frames and Riesz Bases, 2nd ed., Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2016. 10.1007/978-3-319-25613-9_7Search in Google Scholar

[8] O. Christensen and S. S. Goh, Fourier-like frames on locally compact abelian groups, J. Approx. Theory 192 (2015), 82–101. 10.1016/j.jat.2014.11.002Search in Google Scholar

[9] B. Currey, A. Mayeli and V. Oussa, Characterization of shift-invariant spaces on a class of nilpotent Lie groups with applications, J. Fourier Anal. Appl. 20 (2014), no. 2, 384–400. 10.1007/s00041-013-9316-zSearch in Google Scholar

[10] S. Dahlke, Multiresolution analysis and wavelets on locally compact abelian groups, Wavelets, Images, and Surface Fitting (Chamonix-Mont-Blanc 1993), A K Peters, Wellesley (1994), 141–156. Search in Google Scholar

[11] S. R. Das, R. Velsamy and R. Ramakrishnan, Twisted shift-invariant system in L 2 ⁢ ( R 2 ⁢ N ) , Nagoya Math. J. 251 (2023), 734–767. 10.1017/nmj.2023.11Search in Google Scholar

[12] Y. A. Farkov, Orthogonal wavelets with compact supports on locally compact abelian groups, Izv. Ross. Akad. Nauk Ser. Mat. 69 (2005), no. 3, 193–220. 10.4213/im644Search in Google Scholar

[13] G. B. Folland, Harmonic Analysis in Phase Space, Ann. of Math. Stud. 122, Princeton University, Princeton, 1989. 10.1515/9781400882427Search in Google Scholar

[14] G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math., CRC Press, Boca Raton, 1995. Search in Google Scholar

[15] M. Holschneider, Wavelet analysis over abelian groups, Appl. Comput. Harmon. Anal. 2 (1995), no. 1, 52–60. 10.1006/acha.1995.1004Search in Google Scholar

[16] J. W. Iverson, Frames generated by compact group actions, Trans. Amer. Math. Soc. 370 (2018), no. 1, 509–551. 10.1090/tran/6954Search in Google Scholar

[17] M. S. Jakobsen and J. Lemvig, Co-compact Gabor systems on locally compact abelian groups, J. Fourier Anal. Appl. 22 (2016), no. 1, 36–70. 10.1007/s00041-015-9407-0Search in Google Scholar

[18] R. A. Kamyabi Gol and R. R. Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl. 340 (2008), no. 1, 219–225. 10.1016/j.jmaa.2007.08.039Search in Google Scholar

[19] G. Kutyniok and D. Labate, The theory of reproducing systems on locally compact abelian groups, Colloq. Math. 106 (2006), no. 2, 197–220. 10.4064/cm106-2-3Search in Google Scholar

[20] A. Mayeli, Shannon multiresolution analysis on the Heisenberg group, J. Math. Anal. Appl. 348 (2008), no. 2, 671–684. 10.1016/j.jmaa.2008.07.035Search in Google Scholar

[21] R. Radha and S. Adhikari, Frames and Riesz bases of twisted shift-invariant spaces in L 2 ⁢ ( R 2 ⁢ n ) , J. Math. Anal. Appl. 434 (2016), no. 2, 1442–1461. 10.1016/j.jmaa.2015.07.040Search in Google Scholar

[22] R. Radha and S. Adhikari, Shift-invariant spaces with countably many mutually orthogonal generators on the Heisenberg group, Houston J. Math. 46 (2020), no. 2, 435–463. Search in Google Scholar

[23] R. Ramakrishnan and R. Velsamy, Zak transform associated with the Weyl transform and the system of twisted translates on R 2 ⁢ n , Results Math. 79 (2024), no. 2, Paper No. 65. 10.1007/s00025-023-02088-xSearch in Google Scholar

[24] W. Rudin, Fourier Analysis on Groups, Interscience Tracts Pure Appl. Math. 12, Interscience, New York, 1962. Search in Google Scholar

[25] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progr. Math. 159, Birkhäuser, Boston, 1998. 10.1007/978-1-4612-1772-5Search in Google Scholar
Received: 2024-08-12
Published Online: 2025-03-25
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
https://doi.org/10.1515/forum-2024-0385
Keywords for this article
Bessel sequence; frames; Gabor system; Heisenberg group; Hilbert–Schmidt operator; orthonormal system; Parseval frame; Riesz basis; unitary representation; Weyl transform

Articles in the same Issue

  1. Frontmatter
  2. Weil representations of twisted loop groups of type A n (2)
  3. Local Birkhoff decompositions for loop groups and a finiteness result
  4. Uniform boundedness of oscillatory singular integrals with rational phases
  5. On the stabilizer of the graph of linear functions over finite fields
  6. Global solution and blow-up of critical heat equation with nonlocal interaction
  7. Locally compact groups with all dense subgroups separable
  8. On the Kawaguchi–Silverman conjecture for birational automorphisms of irregular varieties
  9. On isometry groups of gradient Ricci solitons
  10. Extensions of a theorem of P. Hall on indexes of maximal subgroups
  11. A trace formula for Hecke operators on Fuchsian groups
  12. Quasi-hereditary algebras with all standard modules being isomorphic to submodules of projective modules
  13. Computation of endo-fixed closures in free-abelian times free groups
  14. Non-vanishing of Maass form 𝐿-functions of cubic level at the central point
  15. Traces of partition Eisenstein series
  16. Gabor System based on the unitary dual of the Heisenberg group
  17. Zeros of linear combinations of Dirichlet 𝐿-functions on the critical line
  18. Paley inequality for the Weyl transform and its applications
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