Journal of the Iranian Mathematical Society

Journal of the Iranian Mathematical Society

Weyl-Zak transforms and twisted translations‎: ‎A representation viewpoint on LCA groups

Document Type : Research Article

Authors
F. Esmaeelzadeh 1
R. A. Kamyabi Gol 2
1 Department of Mathematics, Boj.C., Islamic Azad University, Bojnord, Iran.
2 Department of Mathematics, Center of Excellency in Analysis on Algebric Structures (CEAAS), Ferdowsi University of Mashhad, Mashhad, Iran.
10.30504/jims.2026.572575.1317
Abstract
This paper presents a unified approach to the Weyl-Zak transform on a locally compact abelian (LCA) group $G$‎, ‎grounded in projective representation theory of the time-frequency plane $G \times \widehat{G}$‎. ‎We define a projective representation and a twisted translation on this plane and prove their central intertwining relation with the Weyl transform‎. ‎Using a uniform lattice $L \subset G$ and its annihilator $L^\perp \subset \widehat{G}$‎, ‎we derive a generalized Weyl-Zak transform and show it is unitary from $L^2(G \times \widehat{G})$ to $L^2(S_L \times S_{L^\perp} \times G)$‎, ‎where $S_L$ and $S_{L^\perp}$ are fundamental domains for $L$ and $L^\perp$‎. ‎This provides a cohesive‎, ‎representation-theoretic foundation beyond the Euclidean setting‎. ‎Finally‎, ‎we introduce a bracket product based on the Weyl-Zak transform and demonstrate its essential properties regarding twisted translations.
Keywords
&lrm
Projective representation&lrm
twisted translates&lrm
Weyl transforms&lrm
, &lrm
Weyl-Zak transforms
Subjects
[1] A. Arefijamaal and A. Ghaani Farashahi, Zak transform for semidirect product of locally compact groups, Anal. Math. Phys. 3 (2013), no. 3, 263–276.
[2] P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, Oxford, 1981.
[3] H. G. Feichtinger, W. Kozek, and F. Luef, Gabor analysis over finite Abelian groups, Appl. Comput. Harmon. Anal. 26 (2009), no. 2, 230–248.
[4] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, 1994.
[5] G. B. Folland, Harmonic Analysis in Phase Space, Annals of Mathematics Studies, Princeton University Press, Princeton, 1989.
[6] A. Ghaani Farashahi, Banach convolution modules of group algebras on covariant functions of characters of normal subgroups, Bull. Sci. Math. 185 (2023), Paper No. 103259, 19 pages.
[7] A. Ghaani Farashahi, Generalized Weyl-Heisenberg groups, Anal. Math. Phys. 4 (2014), no. 3, 187–197.
[8] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups, Bull. Malays. Math. Sci. Soc. 38 (2015), no. 2, 779–803.
[9] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transforms over locally compact Abelian groups, Banach J. Math. Anal. 11 (2017), no. 1, 50–71.
[10] A. Ghaani Farashahi, Abstract relative Gabor transforms over canonical homogeneous spaces of semidirect product groups with Abelian normal factor, Anal. Appl. (Singap.) 15 (2017), no. 6, 795–813.
[11] A. Ghaani Farashahi, Covariant function algebras of invariant characters of normal subgroups, J. Geom. Anal. (2026), no. 6, Paper No. 224, 38 pages.
[12] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001.
[13] K. Gröchenig, Aspects of Gabor analysis on locally compact abelian groups, in H. G. Feichtinger and T. Strohmer (eds.), Gabor Analysis and Algorithms, 211–231, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 1998.
[14] A. Kleppner, Multipliers on abelian groups, Math. Ann. 158 (1965) 11–34.
[15] A. Kleppner, Continuity and measurability of multiplier and projective representations, J. Funct. Anal. 17 (1974) 214–226.
[16] G. Kutyniok , The Zak transform on certain locally compact groups, J. Math. Sci. 1 (2002) 62–85.
[17] H. Leptin, Verallgemeinerte L1-Algebren und projektive Darstellungen lokal kompakter Gruppen I, Invent. Math. 3 (1967) 257–281.
[18] H. Leptin, Verallgemeinerte L1-Algebren und projektive Darstellungen lokal kompakter Gruppen II, Invent. Math. 4 (1967) 68–86.
[19] H. Leptin, Darstellungen verallgemeinerter L1-Algebren, Invent. Math. 5 (1968) 192–215.
[20] L. Peng and J. Zhao, Weyl transforms associated with the Heisenberg group, Bull. Sci. Math. 132 (2008), no. 1, 78–86.
[21] R. Radha and S. Adhikari, Frames and Riesz bases of twisted shift-invariant spaces in L2(R2n), J. Math. Anal. Appl. 434 (2016), no. 2, 1442–1461.
[22] R. Radha and N. Shravan Kumar, Weyl transform and Weyl multipliers associated with locally compact abelian groups, J. Pseudo-Differ. Oper. Appl. 9 (2018), no. 2, 229–245.
[23] R. Ramakrishnan and R. Velsamy, Zak transform associated with the Weyl transform and the system of twisted translates on R2n, Results Math. 79 (2024), no. 2, Paper No. 65, 26 pages.
[24] M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, 1981.
[25] M. W. Wong, Wavelet Transform and Localization Operator, Birkhäuser Verlag, Basel, 2002.
Journal of the Iranian Mathematical Society
Volume 7, Issue 2
This issue is in progress but all papers are fully citable
August 2026
Pages 139-153