Document Type : Research Paper
Authors
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.
Abstract
In this paper we consider (extended) metaplectic representation of the semidirect product $G_{\mathbb{J}}=\mathbb{R}^{2d}\times\mathbb{J}$ where $\mathbb{J}$ is a closed subgroup of $Sp(d,\mathbb{R})$, the symplectic group. We will investigate continuous representation frame on $G_{\mathbb{J}}$. We also discuss the existence of duals for such frames and give several characterization for them. Finally, we rewrite the dual conditions, by using the Wigner distribution and obtain more reconstruction formulas.
Keywords
[1] S.T. Ali, J.P. Antoine and J.P. Gazeau, Coherent states, wavelets and their generalizations, Springer-Verlag, New York, 2000.
[2] S.T. Ali, J.P. Antoine and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann. Physics, 222 (1993), pp. 1-37.
[3] J.P. Antoine, The continuous wavelet transform in image processing, CWI Quarterly, 1 (1998), pp. 323-346.
[4] A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math. 37 (2013), pp. 71-79.
[5] P.G. Casazza, G. Kutyniok and M.C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004), pp. 383-408.
[6] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston, 2008.
[7] O. Christensen and S.S. Goh, From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa, Appl. Comput. Harmon. Anal., 36 (2014), pp. 198-214.
[8] E. Cordero, F.D. Mari, K. Nowak and A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. appl. 12 (2006), pp. 157-180.
[9] E. Cordero, E.D. Mari, K. Nowak and A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nachr. 283 (2010), pp. 982-993.
[10] E. Cordero and A. Tabacco, Triangular Subgroups of Sp(d;R) and Reproducing Formulae, J. Funct. Anal. 264 (2013), pp. 2034-2058.
[11] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), pp. 961-1005.
[12] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process, 14 (2005), pp. 2091-2016.
[13] M. Duval-Destin, M.A. Muschietti and B. Torresani, Continuous wavelet decompositions, multiresolution and contrast analysis, SIAM J. Math. Anal. 24 (1993), pp. 739-755.
[14] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
[15] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press: Boca Raton, 1995.
[16] M. Frazier, G. Garrigos, K. Wang and G. Weiss, A characterization of functions that generate wavelet and related expansions, J. Fourier Anal. Appl. 3 (1997), 883-906.
[17] A. Ghaani Farashahi, Square-integrability of multivariate metaplectic wave-packet representations, J. Phys. A, 50 (2017), pp. 115-202.
[18] A. Ghaani Farashahi, Square-integrability of metaplectic wave-packet representations on $L^2(mathbb{R})$}, J. Math. Anal. Appl., 449 (2017), pp. 769-792.
[19] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transforms over locally compact abelian groups, Banach J. Math. Anal., 11 (2017), pp. 50-71.
[20] A. Ghaani Farashahi, Multivariate wave-packet transforms, J. Anal. Appl., 36 (2017), pp. 481-500.
[21] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489 (2016), pp. 75-92.
[22] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), pp. 507-529.
[23] D. Gosson, Symplectic Geometry and Quantum Mechanics, Birkhauser, Basel, 2006.
[24] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
[25] D. Han, Frame representations and parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc., 360 (2008), pp. 3307-3326.
[26] G. Kaiser, A Firendly Guide to wavelets, Birkhauser, Boston, 1994.
[27] V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981.
[2] S.T. Ali, J.P. Antoine and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann. Physics, 222 (1993), pp. 1-37.
[3] J.P. Antoine, The continuous wavelet transform in image processing, CWI Quarterly, 1 (1998), pp. 323-346.
[4] A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math. 37 (2013), pp. 71-79.
[5] P.G. Casazza, G. Kutyniok and M.C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004), pp. 383-408.
[6] O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston, 2008.
[7] O. Christensen and S.S. Goh, From dual pairs of Gabor frames to dual pairs of wavelet frames and vice versa, Appl. Comput. Harmon. Anal., 36 (2014), pp. 198-214.
[8] E. Cordero, F.D. Mari, K. Nowak and A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. appl. 12 (2006), pp. 157-180.
[9] E. Cordero, E.D. Mari, K. Nowak and A. Tabacco, Dimensional upper bounds for admissible subgroups for the metaplectic representation, Math. Nachr. 283 (2010), pp. 982-993.
[10] E. Cordero and A. Tabacco, Triangular Subgroups of Sp(d;R) and Reproducing Formulae, J. Funct. Anal. 264 (2013), pp. 2034-2058.
[11] I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), pp. 961-1005.
[12] M.N. Do and M. Vetterli, The contourlet transform: an efficient directional multiresolution image representation, IEEE Trans. Image Process, 14 (2005), pp. 2091-2016.
[13] M. Duval-Destin, M.A. Muschietti and B. Torresani, Continuous wavelet decompositions, multiresolution and contrast analysis, SIAM J. Math. Anal. 24 (1993), pp. 739-755.
[14] G.B. Folland, Harmonic Analysis in Phase Space, Princeton University Press, 1989.
[15] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press: Boca Raton, 1995.
[16] M. Frazier, G. Garrigos, K. Wang and G. Weiss, A characterization of functions that generate wavelet and related expansions, J. Fourier Anal. Appl. 3 (1997), 883-906.
[17] A. Ghaani Farashahi, Square-integrability of multivariate metaplectic wave-packet representations, J. Phys. A, 50 (2017), pp. 115-202.
[18] A. Ghaani Farashahi, Square-integrability of metaplectic wave-packet representations on $L^2(mathbb{R})$}, J. Math. Anal. Appl., 449 (2017), pp. 769-792.
[19] A. Ghaani Farashahi, Abstract harmonic analysis of wave packet transforms over locally compact abelian groups, Banach J. Math. Anal., 11 (2017), pp. 50-71.
[20] A. Ghaani Farashahi, Multivariate wave-packet transforms, J. Anal. Appl., 36 (2017), pp. 481-500.
[21] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489 (2016), pp. 75-92.
[22] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), pp. 507-529.
[23] D. Gosson, Symplectic Geometry and Quantum Mechanics, Birkhauser, Basel, 2006.
[24] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston, 2001.
[25] D. Han, Frame representations and parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc., 360 (2008), pp. 3307-3326.
[26] G. Kaiser, A Firendly Guide to wavelets, Birkhauser, Boston, 1994.
[27] V.P. Maslov and M.V. Fedoriuk, Semi-Classical Approximations in Quantum Mechanics, Reidel, Boston, 1981.
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