Existence of representation frames based on wave packet groups

Year 2020, Volume: 49 Issue: 5, 1825 - 1842, 06.10.2020

Ali Akbar Arefijamaal Atefe Razghandi

https://doi.org/10.15672/hujms.540946

Abstract

Let $H$ be a locally compact group, $K$ a locally compact abelian group with dual group $\hat{K}$. In this article, we consider the wave packet group $G_{\Theta}$ which is the semidirect product of locally compact groups $H$ and $K\times \hat{K}$, where $\Theta$ is a continuous homomorphism from $H$ into $Aut(K\times\hat{K})$. We review the quasi regular representation on $G_{\Theta}$ and extend the continuous Zak transform to $L^{2}(G_{\Theta})$. Moreover, we state a continuous frame based on $G_{\Theta}$ to reconstruct the element of $L^{2}\left(K\times \hat{K}\right)$. These results are extended to more general wave packet groups. Finally, we establish some methods to find dual of such continuous frames in the form of original frames.

Keywords

semidirect product groups, quasi regular representation, wave packet groups

References

• [1] S.T. Ali, J.P. Antoine and J.P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000.
• [2] F. Andersson, M. Carlsson and L. Tenorio, On the representation of functions with Gaussian wave packets, J. Fourier Anal. Appl. 18, 146-181, 2012.
• [3] A. Arefijamaal, The continuous Zak transform and generalized Gabor frames, Mediterr. J. Math. Phys. 10 (1), 353-365, 2013.
• [4] A. Arefijamaal and A. Ghaani Farashahi, Zak transform for semidirect product of locally compact groups, Anal. Math. Phys. 3 (3), 263-276, 2013.
• [5] A. Arefijamaal and R.A. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom Anal. 19 (3), 541-552, 2009.
• [6] O. Christensen, Pairs of dual Gabor frame generators with compact support and desired frequency localization, Appl. Comput. Harmon. Anal. 20 (3), 403-410, 2006.
• [7] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.
• [8] C.K. Chui and X. Shi, Orthonormal wavelets and tight frames with arbitrary real dilation, Appl. Comput. Harmon. Anal. 9 (3), 243-264, 2000.
• [9] A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Part. Diff. Equat. 3 (11), 979-1005, 1978.
• [10] I. Daubechies, The wavelet transform, time frequency locallization and signal analysis, IEEE Trans. Inform. Theory. 36 (5), 961-1005, 1990.
• [11] I. Daubechies and B. Han, The canonical dual frame of a wavelet frame, Harmon. Anal. 12, 269-285, 2002.
• [12] I. Daubechies and B. Han, Pairs of dual wavelet frames from any two refinable functions, Constr. Approx. 20, 325-352, 2004.
• [13] J. Epperson, Hermite and Laguerre wave packet expansions, Studia Math. 126 (3), 199-217, 1998.
• [14] G.B. Folland, A Course in Abstract Harmonic Analysis, CRCPress, Boca Raton, 1995.
• [15] I.M. Gelfand, Eigen function expansions for equations with periodic coefficients, Dokl. Akad. Nauk. SSR 73, 1117-1120, 1950.
• [16] A. Ghaani Farashahi, Generalized Weyl-Heisenberg groups, Anal. Math. Phys. 4 (3), 187-197, 2014.
• [17] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transforms over locally compact abelian groups, Anal. Math. Banach. J. 11, 50-71, 2017.
• [18] A. Ghaani Farashahi, Square-integrability of metaplectic wave packet representation on $L^{2}\left(\mathbb{R}\right)$, J. Math. Anal. Appl. 449, 769-92, 2017.
• [19] A. Ghaani Farashahi, Theoretical frame properties of wave-packet matrices over prime fields, Linear Multilinear Algebra 11, 2017.
• [20] A. Ghaani Farashahi, Square-integrability of multivariate metaplectic wave-packet representations, J. Phys. A 50, 115-202, 2017.
• [21] A. Ghaani Farashahi, Multivariate wave-packet transforms, Z. Anal. Anwend. 36 (4), 481-500, 2017.
• [22] A. Ghaani Farashahi, Abstract coherent state transforms over homogeneous spaces of compact groups, Complex Anal. Oper. Theory 12, 15-37, 2018.
• [23] K. Gröchenig, Aspects of Gabor analysis on locally compact Abelian groups, in: Gabor Analysis and Algorithms, Birkhäuser Boston, 211-231, 1998.
• [24] K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
• [25] E. Hernandez, D. Labate and G. Weiss, A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal. 12 (4), 615-662, 2002.
• [26] E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Springer-Verlag, Berlin, Vol I, 1963.
• [27] A.J.E.M. Janssen, The Zak transform: a Signal transform for sampled timecontinuous signals Philips J. Res. 43, 23-69, 1988.
• [28] E. Kaniuth and G. Kutyniok, Zeros of the Zak transforms on locally compact abelian groups, Proc. Amer. Math. Soc. 126, 3561-3569, 1998.
• [29] T.H. Koornwinder, Wavelets: An Elementary Treatment of Theory and Applications, World Scientific, Singapore, (1993).
• [30] G. Kutyniok, A qualitative uncertainty principle for functions generating a Gabor frame on LCA groups, J. Math. Anal. Appl. 279, 580-596, 2003.
• [31] D. Labate, G. Weiss and E. Wilson, An approach to the study of wave packet systems, wavelet, frames and operator theory, Contemporary Mathematics 345, 215-235, 2004.
• [32] J. Lemvig, Constructing pairs of dual bandlimited framelets with desired time localization, Adv. Comput Math. 30, 231-247, 2009.
• [33] V. Runde, Lectures on Amenability, Springer, Berlin, 2002.
• [34] J. Zak, Finite translations in solid state physics, Phys. Rev. lett. 19, 1967.