Abstract
An interesting question about the perturbed sequences is: when do they inherit the properties of the original one? An elegant relation between frames (fusion frames) and their perturbations is the relation of their redundancies. In this paper, we investigate these relationships. Also, we express the redundancy of frames (fusion frames) in terms of the cosine angle between some subspaces.
Acknowledgements
The authors would like to thank referee(s) for valuable comments and suggestions.
References
[1] R. Balan, Stability theorems for Fourier frames and wavelet Riesz bases, J. Fourier Anal. Appl. 3 (1997), no. 5, 499–504. 10.1007/BF02648880Search in Google Scholar
[2] B. G. Bodmann, P. G. Casazza and G. Kutyniok, A quantitative notion of redundancy for finite frames, Appl. Comput. Harmon. Anal. 30 (2011), no. 3, 348–362. 10.1016/j.acha.2010.09.004Search in Google Scholar
[3] A. M. Bruckstein, D. L. Donoho and M. Elad, From sparse solutions of systems of equations to sparse modeling of signals and images, SIAM Rev. 51 (2009), no. 1, 34–81. 10.1137/060657704Search in Google Scholar
[4] P. G. Casazza and G. Kutyniok, Frames of subspaces, Wavelets, Frames and Operator Theory, Contemp. Math. 345, American Mathematical Society, Providence (2004), 87–113. 10.1090/conm/345/06242Search in Google Scholar
[5] P. Casazza and G. Kutyniok, Finite Frames: Theory and Applications, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2013. 10.1007/978-0-8176-8373-3Search in Google Scholar
[6] P. G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal. 25 (2008), no. 1, 114–132. 10.1016/j.acha.2007.10.001Search in Google Scholar
[7] P. G. Casazza and J. C. Tremain, The Kadison–Singer problem in mathematics and engineering, Proc. Natl. Acad. Sci. USA 103 (2006), no. 7, 2032–2039. 10.1073/pnas.0507888103Search in Google Scholar PubMed PubMed Central
[8] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, 2003. 10.1007/978-0-8176-8224-8Search in Google Scholar
[9] O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997), 33–47. 10.1002/mana.3211850104Search in Google Scholar
[10] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. 10.1090/S0002-9947-1952-0047179-6Search in Google Scholar
[11] Y. C. Eldar and G. D. Forney, Jr., Optimal tight frames and quantum measurement, IEEE Trans. Inform. Theory 48 (2002), no. 3, 599–610. 10.1109/18.985949Search in Google Scholar
[12] A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 6, Article ID 1450041. 10.1142/S0219691314500416Search in Google Scholar
[13] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra 30 (2015), 507–529. 10.13001/1081-3810.2903Search in Google Scholar
[14] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl. 489 (2016), 75–92. 10.1016/j.laa.2015.10.001Search in Google Scholar
[15] S. Heineken, E. Mathusika and V. Paternostro, Perturbed frame sequences: canonical dual systems, approximate reconstructions and applications, Int. J. Wavelets Multiresolut. Inf. Process. 12 (2014), no. 2, Article ID 1450019. 10.1142/S0219691314500192Search in Google Scholar
[16] V. Kaftal, D. R. Larson and S. Zhang, Operator-valued frames, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6349–6385. 10.1090/S0002-9947-09-04915-0Search in Google Scholar
[17] G. Kutyniok, J. Lemvig and W. Q. Lim, Optimally sparse approximations of 3D functions by compactly supported shearlet frames, SIAM J. Math. Anal. 44 (2012), no. 4, 2962–3017. 10.1137/110844726Search in Google Scholar
[18] G. Kutyniok, V. Mehrmann and P. Petersen, Regularization and numerical solution of the inverse scattering problem using shearlet frames, J. Inverse Ill-Posed Probl. 25 (2017), no. 3, 287–309. 10.1515/jiip-2015-0048Search in Google Scholar
[19] L. Oeding, E. Robeva and B. Sturmfels, Decomposing tensors into frames, Adv. in Appl. Math. 73 (2016), 125–153. 10.1016/j.aam.2015.10.004Search in Google Scholar
[20] A. Rahimi, G. Zandi and B. Daraby, Redundancy of fusion frames in Hilbert spaces, Complex Anal. Oper. Theory 10 (2016), no. 3, 545–565. 10.1007/s11785-015-0489-0Search in Google Scholar
[21] W. Sun, g-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), no. 1, 437–452. 10.1016/j.jmaa.2005.09.039Search in Google Scholar
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