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Systems of left translates and oblique duals on the Heisenberg group

Year 2023, , 222 - 236, 15.12.2023
https://doi.org/10.33205/cma.1382306

Abstract

In this paper, we characterize the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$, $g\in L^2(\mathbb{H})$, to be a frame sequence or a Riesz sequence in terms of the twisted translates of the corresponding function $g^\lambda$. Here, $\mathbb{H}$ denotes the Heisenberg group and $g^\lambda$ the inverse Fourier transform of $g$ with respect to the central variable. This type of characterization for a Riesz sequence allows us to find some concrete examples. We also study the structure of the oblique dual of the system of left translates $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ on $\mathbb{H}$. This result is also illustrated with an example.

References

  • M. Abramowitz, I. Stegun: Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards (1972).
  • S. Arati, R. Radha: Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indag. Math. (N.S.), 30 (1) (2019), 106–127.
  • S. Arati, R. Radha: Orthonormality of wavelet system on the Heisenberg group, J. Math. Pures Appl., 131 (2019), 171–192.
  • S. Arati, R. Radha: Wavelet system and Muckenhoupt $A_2$ condition on the Heisenberg group, Colloq. Math., 158 (1) (2019), 59–76.
  • D. Barbieri, E. Hernández, and A. Mayeli: Bracket map for the Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal., 37 (2) (2014), 218–234.
  • M. Bownik: The structure of shift-invariant subspaces of $L^2(R^n)$, J. Funct. Anal., 177 (2) (2000), 282–309.
  • M. Bownik, K. A. Ross: The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl., 21 (4) (2015), 849–884.
  • C. Cabrelli, V. Paternostro: Shift-invariant spaces on LCA groups, J. Funct. Anal., 258 (6) (2010), 2034–2059.
  • O. Christensen: An Introduction to Frames and Riesz Bases, second ed., Applied and Numerical Harmonic Analysis, Birkhäuser/Springer [Cham] (2016).
  • O. Christensen, H. O. Kim, R. Y. Kim, and J. K. Lim: Riesz sequences of translates and generalized duals with support on [0, 1], J. Geom. Anal., 16 (4) (2006), 585–596.
  • 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 (2) (2014), 384–400.
  • S. R. Das, R. Radha: Shift-invariant system on the Heisenberg Group, Adv. Oper. Theory, 6 (1) (2021), 27.
  • G. B. Folland: Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ (1989).
  • J. W. Iverson: Frames generated by compact group actions, Trans. Amer. Math. Soc., 370 (1) (2018), 509–551.
  • M. S. Jakobsen, J. Lemvig: Reproducing formulas for generalized translation invariant systems on locally compact abelian groups, Trans. Amer. Math. Soc., 368 (12) (2016), 8447–8480.
  • R. A. Kamyabi Gol, R. R. Tousi: The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340 (1) (2008), 219–225.
  • S. G. Mallat: Multiresolution approximations and wavelet orthonormal bases of $L^2({\bf R})$, Trans. Amer. Math. Soc., 315 (1) (1989), 69–87.
  • Y. Meyer: Ondelettes et fonctions splines, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., Palaiseau, 1987, pp. Exp. No. VI, 18.
  • R. Radha , S. Adhikari: Shift-invariant spaces with countably many mutually orthogonal generators on the Heisenberg group, Houston J. Math., 46 (2) (2020), 435–463.
  • R. Radha, N. S. Kumar: Shift invariant spaces on compact groups, Bull. Sci. Math., 137 (4) (2013), 485–497.
  • S. Thangavelu: Harmonic Analysis on the Heisenberg group, Progress in Mathematics, Vol. 159, Birkhäuser Boston, Inc., Boston, MA (1998).
Year 2023, , 222 - 236, 15.12.2023
https://doi.org/10.33205/cma.1382306

Abstract

References

  • M. Abramowitz, I. Stegun: Handbook of Mathematical Functions, Applied Mathematics Series 55, National Bureau of Standards (1972).
  • S. Arati, R. Radha: Frames and Riesz bases for shift invariant spaces on the abstract Heisenberg group, Indag. Math. (N.S.), 30 (1) (2019), 106–127.
  • S. Arati, R. Radha: Orthonormality of wavelet system on the Heisenberg group, J. Math. Pures Appl., 131 (2019), 171–192.
  • S. Arati, R. Radha: Wavelet system and Muckenhoupt $A_2$ condition on the Heisenberg group, Colloq. Math., 158 (1) (2019), 59–76.
  • D. Barbieri, E. Hernández, and A. Mayeli: Bracket map for the Heisenberg group and the characterization of cyclic subspaces, Appl. Comput. Harmon. Anal., 37 (2) (2014), 218–234.
  • M. Bownik: The structure of shift-invariant subspaces of $L^2(R^n)$, J. Funct. Anal., 177 (2) (2000), 282–309.
  • M. Bownik, K. A. Ross: The structure of translation-invariant spaces on locally compact abelian groups, J. Fourier Anal. Appl., 21 (4) (2015), 849–884.
  • C. Cabrelli, V. Paternostro: Shift-invariant spaces on LCA groups, J. Funct. Anal., 258 (6) (2010), 2034–2059.
  • O. Christensen: An Introduction to Frames and Riesz Bases, second ed., Applied and Numerical Harmonic Analysis, Birkhäuser/Springer [Cham] (2016).
  • O. Christensen, H. O. Kim, R. Y. Kim, and J. K. Lim: Riesz sequences of translates and generalized duals with support on [0, 1], J. Geom. Anal., 16 (4) (2006), 585–596.
  • 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 (2) (2014), 384–400.
  • S. R. Das, R. Radha: Shift-invariant system on the Heisenberg Group, Adv. Oper. Theory, 6 (1) (2021), 27.
  • G. B. Folland: Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122, Princeton University Press, Princeton, NJ (1989).
  • J. W. Iverson: Frames generated by compact group actions, Trans. Amer. Math. Soc., 370 (1) (2018), 509–551.
  • M. S. Jakobsen, J. Lemvig: Reproducing formulas for generalized translation invariant systems on locally compact abelian groups, Trans. Amer. Math. Soc., 368 (12) (2016), 8447–8480.
  • R. A. Kamyabi Gol, R. R. Tousi: The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340 (1) (2008), 219–225.
  • S. G. Mallat: Multiresolution approximations and wavelet orthonormal bases of $L^2({\bf R})$, Trans. Amer. Math. Soc., 315 (1) (1989), 69–87.
  • Y. Meyer: Ondelettes et fonctions splines, Séminaire sur les équations aux dérivées partielles 1986–1987, École Polytech., Palaiseau, 1987, pp. Exp. No. VI, 18.
  • R. Radha , S. Adhikari: Shift-invariant spaces with countably many mutually orthogonal generators on the Heisenberg group, Houston J. Math., 46 (2) (2020), 435–463.
  • R. Radha, N. S. Kumar: Shift invariant spaces on compact groups, Bull. Sci. Math., 137 (4) (2013), 485–497.
  • S. Thangavelu: Harmonic Analysis on the Heisenberg group, Progress in Mathematics, Vol. 159, Birkhäuser Boston, Inc., Boston, MA (1998).
There are 21 citations in total.

Details

Primary Language English
Subjects Lie Groups, Harmonic and Fourier Analysis
Journal Section Articles
Authors

Santi Das 0000-0002-8553-5973

Radha Ramakrishnan 0000-0001-5576-7960

Peter Massopust 0000-0002-5466-1336

Early Pub Date November 13, 2023
Publication Date December 15, 2023
Submission Date October 29, 2023
Acceptance Date November 9, 2023
Published in Issue Year 2023

Cite

APA Das, S., Ramakrishnan, R., & Massopust, P. (2023). Systems of left translates and oblique duals on the Heisenberg group. Constructive Mathematical Analysis, 6(4), 222-236. https://doi.org/10.33205/cma.1382306
AMA Das S, Ramakrishnan R, Massopust P. Systems of left translates and oblique duals on the Heisenberg group. CMA. December 2023;6(4):222-236. doi:10.33205/cma.1382306
Chicago Das, Santi, Radha Ramakrishnan, and Peter Massopust. “Systems of Left Translates and Oblique Duals on the Heisenberg Group”. Constructive Mathematical Analysis 6, no. 4 (December 2023): 222-36. https://doi.org/10.33205/cma.1382306.
EndNote Das S, Ramakrishnan R, Massopust P (December 1, 2023) Systems of left translates and oblique duals on the Heisenberg group. Constructive Mathematical Analysis 6 4 222–236.
IEEE S. Das, R. Ramakrishnan, and P. Massopust, “Systems of left translates and oblique duals on the Heisenberg group”, CMA, vol. 6, no. 4, pp. 222–236, 2023, doi: 10.33205/cma.1382306.
ISNAD Das, Santi et al. “Systems of Left Translates and Oblique Duals on the Heisenberg Group”. Constructive Mathematical Analysis 6/4 (December 2023), 222-236. https://doi.org/10.33205/cma.1382306.
JAMA Das S, Ramakrishnan R, Massopust P. Systems of left translates and oblique duals on the Heisenberg group. CMA. 2023;6:222–236.
MLA Das, Santi et al. “Systems of Left Translates and Oblique Duals on the Heisenberg Group”. Constructive Mathematical Analysis, vol. 6, no. 4, 2023, pp. 222-36, doi:10.33205/cma.1382306.
Vancouver Das S, Ramakrishnan R, Massopust P. Systems of left translates and oblique duals on the Heisenberg group. CMA. 2023;6(4):222-36.