Ali Akbar AREFİJAMAAL, Atefe Razghandi

Existence of representation frames based on wave packet groups

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,

PDF

___

[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.