g-frame generator sets for projective unitary representations

Aifang Liu; Jian Wu; Aifang Liu; Jian Wu

doi:10.3934/math.2024800

AIMS Mathematics

2024, Volume 9, Issue 6: 16506-16525. doi: 10.3934/math.2024800

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Research article

g-frame generator sets for projective unitary representations

Aifang Liu ^,,
Jian Wu

College of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China

Received: 17 March 2024 Revised: 24 April 2024 Accepted: 08 May 2024 Published: 13 May 2024
MSC : 42C15, 42C40, 47D03

Frames with special structures play a crucial role in various industrial applications, such as medical imaging, quantum communication, recognition and identification software, and so on. In this paper, we will discuss a more general setting, i.e., a g-frame induced by the projective unitary representation. We show some new results on the dilation property for g-frame generator sets for unitary groups and projective unitary representations. In particular, by using complete wandering operators, several properties of g-frame generators for projective unitary representations have been obtained. Moreover, we explore some characterizations of the g-frame generator dual pairs.
- g-frame generator set,
- projective unitary representation,
- dilation,
- unitary group,
- dual g-frame generator
Citation: Aifang Liu, Jian Wu. g-frame generator sets for projective unitary representations[J]. AIMS Mathematics, 2024, 9(6): 16506-16525. doi: 10.3934/math.2024800

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Abstract

Frames with special structures play a crucial role in various industrial applications, such as medical imaging, quantum communication, recognition and identification software, and so on. In this paper, we will discuss a more general setting, i.e., a g-frame induced by the projective unitary representation. We show some new results on the dilation property for g-frame generator sets for unitary groups and projective unitary representations. In particular, by using complete wandering operators, several properties of g-frame generators for projective unitary representations have been obtained. Moreover, we explore some characterizations of the g-frame generator dual pairs.

References

[1]	R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier serier, Trans. Amer. Math. Soc., 72 (1952), 341–366.
[2]	I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), 1271–1283. https://doi.org/10.1063/1.527388 doi: 10.1063/1.527388
[3]	A. Aldroubi, C. Cabrelli, U. M. Molter, Wavelets on irregular grids with arbitrary dilation matrices and frame atomics for $L^2(\mathbb{R}^d)$, Appl. Comput. Harmon. Anal., 17 (2004), 119–140. https://doi.org/10.1016/j.acha.2004.03.005 doi: 10.1016/j.acha.2004.03.005
[4]	P. G. Casazza, G. Kutyniok, Frames of subspaces, wavelets, frames and operator theory, Contemp. Math., Amer. Math. Soc., 139 (2004), 87–113.
[5]	O. Christensen, Y. C. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comput. Harmon. Anal., 17 (2004), 48–68. https://doi.org/10.1016/j.acha.2003.12.003 doi: 10.1016/j.acha.2003.12.003
[6]	S. Li, H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl., 10 (2004), 409–431. https://doi.org/10.1007/s00041-004-3039-0 doi: 10.1007/s00041-004-3039-0
[7]	W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322 (2006), 437–452. https://doi.org/10.1016/j.jmaa.2005.09.039 doi: 10.1016/j.jmaa.2005.09.039
[8]	M. R. Abdollahpour, Dilation of dual g-frames to dual g-Riesz bases, Banach J. Math. Anal., 9 (2015), 54–66. http://doi.org/10.15352/bjma/09-1-5 doi: 10.15352/bjma/09-1-5
[9]	A. Arefijamaal, S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math., 37 (2013), 71–79. https://doi.org/10.3906/mat-1107-14 doi: 10.3906/mat-1107-14
[10]	M. Bownik, J. Jasper, D. Speegle, Orthonormal dilations of non-tight frames, Proc. Amer. Math. Soc., 139 (2011), 3247–3256. https://doi.org/10.48550/arXiv.1004.3552 doi: 10.48550/arXiv.1004.3552
[11]	S. Hosseini, A. Khosravi, G-frames and operator-valued frames in Hilbert spaces, Int. Math. Forum, 5 (2010), 1597–1606.
[12]	V. Kaftal, D. Larson, S. Zhang, Operator-valued frames, Trans. Amer. Math. Soc., 361 (2009), 6349–6385.
[13]	D. Han, Approximations for Gabor and wavelet frames, Trans. Amer. Math. Soc., 355 (2003), 3329–3342. https://doi.org/10.1090/S0002-9947-03-03047-2 doi: 10.1090/S0002-9947-03-03047-2
[14]	D. Han, Frame representations and Parseval duals with applications to Gabor frames, Trans. Amer. Math. Soc., 360 (2008), 3307–3326. https://doi.org/10.1090/S0002-9947-08-04435-8 doi: 10.1090/S0002-9947-08-04435-8
[15]	D. Han, D. R. Larson, Frames, bases and group representations, 2000.
[16]	A. Najati, M. H. Faroughi, A. Rahimi, G-frame and stability of G-frames in Hilbert spaces, Methods Funct. Anal. Topol., 14 (2008), 271–286.
[17]	X. Dai, D. R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs of the American Mathematical Society, Vol. 134, 1998. https://doi.org/10.1090/memo/0640
[18]	B. Blackadar, Operator algebras: theory of $C^$-algebras and von Neumann algebras*, Berlin: Springer, 2006. https://doi.org/10.1007/3-540-28517-2
[19]	K. Musazadeh, A. Khosravi, G-frame generator sets, 41th Iranian International Conference on Mathematics, 2010.
[20]	D. Han, D. Larson, Frame duality properties for projective unitary representations, B. Lond. Math. Soc., 40 (2008), 685–695. https://doi.org/10.1112/blms/bdn049 doi: 10.1112/blms/bdn049
[21]	X. Guo, Wandering operators for unitary systems of Hilbert spaces, Complex Anal. Oper. Theory, 10 (2015), 703–723. https://doi.org/10.1007/s11785-015-0448-9 doi: 10.1007/s11785-015-0448-9

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