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Gabor System based on the unitary dual of the Heisenberg group

  • Santi R. Das und Radha Ramakrishnan EMAIL logo
Veröffentlicht/Copyright: 25. März 2025
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 37 Heft 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

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Received: 2024-08-12
Published Online: 2025-03-25
Published in Print: 2025-06-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  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
Schlagwörter für diesen Artikel
Bessel sequence; frames; Gabor system; Heisenberg group; Hilbert–Schmidt operator; orthonormal system; Parseval frame; Riesz basis; unitary representation; Weyl transform

Artikel in diesem Heft

  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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