Acta mathematica scientia, Series B >
APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES
Received date: 2012-06-19
Revised date: 2013-07-23
Online published: 2014-05-20
In this article, we introduce and characterize approximate duality for g-frames. We get some important properties and applications of approximate duals. We also obtain some new results in approximate duality of frames, and generalize some of the known results in approximate duality of frames to g-frames. We also get some results for fusion frames, and perturbation of approximately dual g-frames. We show that approximate duals are stable under small perturbations and they are useful for erasures and reconstruction.
Key words： Frame; g-frame; approximate duality; perturbation; reconstruction
Amir KHOSRAVI , Morteza MIRZAEE AZANDARYANI . APPROXIMATE DUALITY OF g-FRAMES IN HILBERT SPACES[J]. Acta mathematica scientia, Series B, 2014 , 34(3) : 639 -652 . DOI: 10.1016/S0252-9602(14)60036-9
[1] Asgari M S. Khosravi A. Frames and bases of subspaces in Hilbert spaces. J Math Anal Appl, 2005, 308: 541–553
[2] Balazs P, Feichtinger H G, Hampejs M, Kracher G. Double preconditioning for Gabor frames. IEEE Trans Signal Process, 2006, 54: 4597–4610
[3] Bourouihiya A. The tensor product of frames. Sampling Theory in Signal and Image Process, 2008, 7(1): 65–76
[4] Casazza P, Balan R, Edidin D, Kutyniok G. Decompositions of frames and a new frame identity. SPIE Wavelets Applications in Signal and Image Processing XI, 2005, 5914: 379–388
[5] Casazza P, Kutyniok G. Frames of subspaces. Contemp Math Amer Math Soc, 2004, 345: 87–113
[6] Casazza P, Kutyniok G. Robustness of fusion frames under erasures of subspaces and of local frame vectors.
Contemp Math Amer Math Soc, 2008, 464: 149–160
[7] Casazza P, Kutyniok G, Li S. Fusion frames and distributed processing. Appl Comput Harmon Anal, 2008, 25: 114–132
[8] Christensen O. Frames and bases. An introductory course. Boston: Birkh¨auser, 2008
[9] Christensen O, Laugesen R S. Approximate dual frames in Hilbert spaces and applications to Gabor frames.
Sampling Theory in Signal and Image Processing, 2011, 9: 77–90
[10] Daubechies I, Grossmann A, Meyer Y. Painless nonorthogonal expansions. J Math Phys, 1986, 27: 1271–1283
[11] Duffin R J, Schaeffer A C. A class of nonharmonic Fourier series. Trans Amer Math Soc, 1952, 72: 341–366
[12] Gavruta P. On the duality of fusion frames. J Math Anal Appl, 2007, 333: 871–879
[13] Gilbert J E, Han Y S, Hogan J A, et al. Smooth molecular decompositions of functions and singular integral
operators. Mem Amer Math Soc, 2002, 742: 1–74
[14] Khosravi A, Khosravi B. Frames and bases in tensor products of Hilbert spaces and Hilbert C-modules. Proc Indian Acad Sci (Math Sci), 2007, 117 (1): 1–12
[15] Khosravi A, Khosravi B. Fusion frames and g-frames in Hilbert C-modules. Int J Wavelets Multiresolut Inf Process, 2008, 6: 433–446
[16] Khosravi A, Khosravi B. Fusion frames and g-frames in Banach spaces. Proc Indian Acad Sci (Math Sci), 2011, 121 (2): 155–164
[17] Khosravi A, Mirzaee Azandaryani M. G-frames and direct sums. Bull Malays Math Soc (2), 2013, 36 (2): 313–323
[18] Khosravi A, Musazadeh K. Fusion frames and g-frames. J Math Anal Appl, 2008, 342: 1068–1083
[19] Murphy G J. C-algebra and operator theory. San Diego: Academic Press, 1990
[20] Obeidat S, Samarah S, Casazza P, Tremain J C. Sums of Hilbert space frames. J Math Anal Appl, 2009, 351: 579–585
[21] Sun W. G-frames and g-Riesz bases. J Math Anal Appl, 2006, 322: 437–452
[22] Werther T, Eldar Y C, Subbanna N K. Dual Gabor frames: theory and computational aspects. IEEE Trans Signal Process, 2005, 53: 4147–4158
[23] Xiao X C, Zeng X M. Some equalities and inequalities of g-continuous frames. Sci China Ser A, 2010, 53(10): 2621–2632
[24] Zhu Y C. Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Math Sinica, English Series, 2008, 24 (10): 1727–1736
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