Document Type: Research Paper

Authors

1 Department of Sciences, Urmia University of Technology, P.O.Box 419-57155, Urmia, Iran.

2 Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.

Abstract

Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual of each other in general. In this paper, we investigate the structure of the duals of fusion frames and discuss the relation between the duals of fusion frames with their associated discrete frames.

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Main Subjects

[1] Z. Amiri, M.A. Dehghan, and E. Rahimi, Subfusion frames, Abstr. Appl. Anal., 2012 (2012) 1-12.

[2] A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math., 37 (2013) 71-79.

[3] A. Arefijamaal, E. Zekaee, Image processing by alternate dual Gabor frames, To appear in Bull. Iranian Math. Soc.

[4] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35 (2013) 535-540.

[5] R. Calderbank, P.G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki, Sparse fusion frames: existence and construction, Adv. Comput. Math., 35  (1) (2011) 1-31.

[6] P.G. Casazza and M. Fickus, Minimizing fusion frame potential, Acta, Appl. Math., 107 (2009) 7-24.

[7] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, (2004) 87-113.

[8] P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25 (1) (2008) 114-132.

[9] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.

[10] O. Christensen and E. Osgooei, On frame properties for Fourier-like systems, J. Approx. Theory, 172 (2013) 47-57.

[11] M.A. Dehghan and M.A. Hasankhani Fard, G-dual frames in Hilbert spaces, U. P. B. Sci. Bull. Series A, 75 (1) (2013) 129-140.

[12] P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl., 333 ( 2) (2007) 871-879.

[13] W.H. Greub, Linear Algebra, Springer-Verlag, New York, 1981.

[14] S.K. Kaushik, A generalization of frames in Banach spaces, J. Contemp. Math. Anal., 44 (4) (2009) 212-218.

[15] J. Leng, Q. Guo, and T. Huang, The duals of fusion frames for experimental data transmission coding of high energy physics, Adv. High Energy Phys., 2013 (2013) 1-9.

[16] P.G. Massey, M.A. Ruiz, and D. Stojanoff, The structure of minimizers of the frame potential on fusion frames, J. Fourier Anal. Appl., 16 (2010) 514-543.

[17] A. Najati, A. Rahimi, and M.H. Faroughi, Continuous and discrete frames of subspaces in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008) 305-324.

[18] E. Osgooei and M.H. Faroughi, Hilbert-Schmidt sequences and dual of g-frames, Acta Univ. Apulensis, 36 (2013) 1-15.

[19] A. Rahimi, Invariance of Fréchet frames under perturbation, Sahand Communications in Mathematical Analysis, 1 (1) (2014), 41-51.