Banach Journal of Mathematical Analysis

Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups

Arash Ghaani Farashahi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article presents a systematic study for abstract harmonic analysis aspects of wave-packet transforms over locally compact abelian (LCA) groups. Let H be a locally compact group, let K be an LCA group, and let θ:HAut(K) be a continuous homomorphism. We introduce the abstract notion of the wave-packet group generated by θ, and we study basic properties of wave-packet groups. Then we study theoretical aspects of wave-packet transforms. Finally, we will illustrate application of these techniques in the case of some well-known examples.

Article information

Banach J. Math. Anal., Volume 11, Number 1 (2017), 50-71.

Received: 9 November 2015
Accepted: 18 February 2016
First available in Project Euclid: 10 November 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 43A30: Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
Secondary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc.

coherent state (covariant) transform wave-packet group wave-packet representation wave-packet transform wavelet (Gabor) transform


Ghaani Farashahi, Arash. Abstract harmonic analysis of wave-packet transforms over locally compact abelian groups. Banach J. Math. Anal. 11 (2017), no. 1, 50--71. doi:10.1215/17358787-3721281.

Export citation


  • [1] S. T. Ali, J. P. Antoine, and J. P. Gazeau, Coherent States, Wavelets and Their Generalizations, 2nd ed., Springer, New York, 2014.
  • [2] A. A. Arefijamaal and R. A. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process 6 (2008), no. 5, 749–759.
  • [3] A. A. Arefijamaal and R. A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal. 19 (2009), no. 3, 541–552.
  • [4] J. J. Benedetto and G. E. Pfander, Periodic wavelet transforms and periodicity detection, SIAM J. Appl. Math. 62 (2002), no. 4, 1329–1368.
  • [5] D. Bernier and K. F. Taylor, Wavelets from square-integrable representations, SIAM J. Math. Anal. 27 (1996), no. 2, 594–608.
  • [6] O. Christensen and A. Rahimi, Frame properties of wave packet systems in $L^{2}(\mathbb{R}^{d})$, Adv. Comput. Math. 29 (2008), no. 2, 101–111.
  • [7] H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal. 86 (1989), no. 2, 307–340.
  • [8] H. G. Feichtinger and K. H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II, Monatsh. Math. 108 (1989), no. 2–3, 129–148.
  • [9] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, Florida, 1995.
  • [10] H. Führ, Abstract Harmonic Analysis of Continuous Wavelet Transforms, Lecture Notes in Math. 1863, Springer, Berlin, 2005.
  • [11] H. Führ and M. Mayer, Continuous wavelet transforms from semidirect products, cyclic representation and Plancherel Measure, J. Fourier Anal. Appl. 8 (2002), no. 4, 375–397.
  • [12] A. Ghaani Farashahi, Cyclic wave-packet transform on finite Abelian groups of prime order, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 6, art ID 1450041.
  • [13] A. Ghaani Farashahi, Wave-packet transform over finite fields, Electron. J. Linear Algebra 30 (2015), 507–529.
  • [14] A. Ghaani Farashahi, Wave-packet transforms over finite cyclic groups, Linear Algebra Appl. 489 (2016), 75–92.
  • [15] K. Gröchenig, “Aspects of Gabor analysis on locally compact Abelian groups” in Gabor Analysis and Algorithms, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 1998, 211–231.
  • [16] A. Grossmann, J. Morlet, and T. Paul, Transforms associated to square integrable group representations, I: General results, J. Math. Phys. 26 (10) (1985), no. 10, 2473–2479.
  • [17] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, I, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963.
  • [18] G. Hochschild, The Structure of Lie Groups, Holden-Day, San Francisco, 1965.
  • [19] C. Kalisa and B. Torrësani, $N$-dimensional affine Weyl-Heisenberg wavelets, Ann. Henri Poincaré A. 59 (1993), no. 2, 201–236.
  • [20] V. Kisil, Wavelets in Banach spaces, Acta Appl. Math. 59 (1999), no. 1, 79–109.
  • [21] V. Kisil, “Wavelets beyond admissibility” in Progress in Analysis and Its Applications (London, 2009), World Scientific, Hackensack, NJ, 2010, 219–225.
  • [22] V. Kisil, “Erlangen program at large: An overview” in Advances in Applied Analysis, Trends Math., Birkhäuser, Basel, 2012, 1–94.
  • [23] V. Kisil, Geometry of Möbius Transformations: Elliptic, Parabolic and Hyperbolic Actions of $SL_{2}(\mathbb{R})$, Imperial College Press, London, 2012.
  • [24] V. Kisil, Operator covariant transform and local principle, J. Phys. A. 45 (2012), no. 24, art ID 244022.
  • [25] V. Kisil, Calculus of operators: Covariant transform and relative convolutions, Banach J. Math. Anal. 8 (2014), no. 2, 156–184.
  • [26] F. Luef and Z. Rahbani, On pseudodifferential operators with symbols in generalized Shubin classes and an application to Landau-Weyl operators, Banach J. Math. Anal. 5 (2011), no. 2, 59–72.
  • [27] G. E. Pfander, “Gabor Frames in Finite Dimensions” in Finite Frames, Appl. Numer. Harmon. Anal., Birkhauser, Boston, 2013, 193–239.
  • [28] B. Torrésani, Wavelets associated with representations of the affine Weyl-Heisenberg group, J. Math. Phys. 32 (1991), no. 5, 1273–1279.
  • [29] B. Torrésani, Time-frequency representation: wavelet packets and optimal decomposition, Ann. Henri Poincaré 56 (1992), no. 2, 215–234.