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Fusion-Riesz frame in Hilbert space

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Abstract

Fusion-Riesz frame (Riesz frame of subspace) whose all subsets are fusion frame sequences with the same bounds is a special fusion frame. It is also considered a generalization of Riesz frame since it shares some important properties of Riesz frame. In this paper, we show a part of these properties of fusion-Riesz frame and the new results about the stabilities of fusion-Riesz frames under operator perturbation (simple named operator perturbation of fusion-Riesz frames). Moreover, we also compare the operator perturbation of fusion-Riesz frame with that of fusion frame, fusion-Riesz basis (also called Riesz decomposition or Riesz fusion basis) and exact fusion frame.

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References

  1. U Ayaz, S Dirksen, H Rauhut. Uniform recovery of fusion frame structured sparse signals, Appl Comput Harmon Anal, 2016, 41: 341–361.

    Article  MathSciNet  MATH  Google Scholar 

  2. M S Asgari. New characterizations of fusion frames (frames of subspaces), Proc Indian Acad Sci Math Sci, 2009, 119(3): 369–382.

    Article  MathSciNet  MATH  Google Scholar 

  3. M S Asgari. On the stability of fusion frames under operators, Math Anal Appl, 2009, 1388: 65–68.

    Google Scholar 

  4. M S Asgari. On the Riesz fusion bases in Hilbert spaces, J Egyptian Math Soc, 2013, 21(2): 79–86.

    Article  MathSciNet  MATH  Google Scholar 

  5. M S Asgari, A Khosravi. Frames and bases of subspaces in Hilbert spaces, J Math Anal Appl, 2005, 308(2): 541–553.

    Article  MathSciNet  MATH  Google Scholar 

  6. H Bölcskei, F Hlawatsch, H G Feichtinger. Frame-theoretic analysis of oversampled filter banks, IEEE Trans Signal Process, 1998, 46: 3256–3268.

    Article  Google Scholar 

  7. P Boufounos, G Kutyniok, H Rauhut. Sparse recovery from combined fusion frame measurements, IEEE Trans Inform Theory, 2011, 57(6): 3864–3876.

    Article  MathSciNet  MATH  Google Scholar 

  8. C Bachoc, M Ehler. Tight p-fusion frames, Appl Comput Harmon Anal, 2013, 35: 1–15.

    Article  MathSciNet  MATH  Google Scholar 

  9. E J Cands, D L Donoho. New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Comm Pure Appl Math, 2004, 56: 216–266.

    Google Scholar 

  10. A Chebira, M Fickus, D G Mixon. Filter bank fusion frames, IEEE Trans Signal Process, 2011, 59: 953–963.

    Article  MathSciNet  Google Scholar 

  11. P G. Casazza, M Fickus, D G Mixon etc. Constructing tight fusion frames, Appl Comput Harmon Anal, 2011, 30: 175–187.

    Article  MathSciNet  MATH  Google Scholar 

  12. P G Casazza, G Kutyniok. Frames of subspaces, Contemp Math, 2004, 345: 87–113.

    Article  MathSciNet  MATH  Google Scholar 

  13. P G Casazza, O Christensen, A M Lindner etc. Frames and the Feichtinger conjecture, Proc Amer Math Soc, 2005, 133(4): 1025–1033.

    Article  MathSciNet  MATH  Google Scholar 

  14. P G Casazza, G Kutyniok, S Li. Fusion frames and distributed processing, Appl Comput Harmon Anal, 2008, 25: 114–132.

    Article  MathSciNet  MATH  Google Scholar 

  15. O Christensen. Frames and Bases: An Introductory Course, Birkhäuser, Boston, M A, 2008.

    Book  MATH  Google Scholar 

  16. O Christensen. Frames containing a Riesz base and approximation of the frame coefficents using finite-dimensional methods, J Math Anal Appl, 1996, 196: 256–270.

    Article  MATH  Google Scholar 

  17. I Daubechies. Ten Lectures on Wavelets, SIAM, Philadelphia, 1992.

    Book  MATH  Google Scholar 

  18. H G Feichtinger, T Strohmer, Eds. Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, MA, 1998.

    Book  MATH  Google Scholar 

  19. P Gavruta. On the duality of fusion frames, J Math Anal Appl, 2007, 333: 871–879.

    Article  MathSciNet  MATH  Google Scholar 

  20. D Han, W Sun. Reconstruction of signals from frame coefficients with erasures at unknown locations, IEEE Trans Inform Theory, 2014, 60: 4013–4025.

    Article  MathSciNet  MATH  Google Scholar 

  21. T Kato, Perturbation Theory for Linear Operator, Springer-Verlag, New York, 1984.

    Google Scholar 

  22. D Larson, S Scholze. Signal reconstruction from frame and sampling erasures, J Fourier Anal Appl, 2015, 21: 1146–1167.

    Article  MathSciNet  MATH  Google Scholar 

  23. F S Lv, W C Sun. Construction of robust frames in erasure recovery, Linear Algebra Appl, 2015, 479: 155–170.

    Article  MathSciNet  MATH  Google Scholar 

  24. X B Li, S Z Yang, Y C Zhu. Some results about operator perturbation of fusion frames in Hilbert spaces, J Math Anal Appl, 2015, 421: 1417–1427.

    Article  MathSciNet  MATH  Google Scholar 

  25. X B Li, Y C Zhu. Operator perturbation of fusion-Riesz frames, J Fuzhou Univ Nat Sci Ed, 2014, 42: 173–176. (In Chinese)

    MathSciNet  Google Scholar 

  26. J Z Li, Y C Zhu. g-Riesz frames in Hilbert spaces, Sci Sin Math, 2011, 41: 53–68.

    Article  Google Scholar 

  27. W Rudin. Functional Analysis, Second Edition, McGraw-Hill, New York, 1991.

    MATH  Google Scholar 

  28. M A Ruiz, D Stojanoff. Some properties of frames of subspaces obtained by operator theory methods, J Math Anal Appl, 2008, 343: 366–378.

    Article  MathSciNet  MATH  Google Scholar 

  29. W Sun. G-frames and g-Riesz bases, J Math Anal Appl, 2006, 322(1): 437–452.

    Article  MathSciNet  MATH  Google Scholar 

  30. J H Wang, Y C Zhu. Some properties of fusion frame, J Fuzhou Univ Nat Sci Ed, 2009, 37(1): 1999–2006. (In Chinese)

    MathSciNet  Google Scholar 

  31. J H Wang, Y C Zhu, Y L Wang. Stabilities of fusion-Riesz frames and g-Riesz frames, J Fuzhou Univ Nat Sci Ed, 2011, 39(4): 469–472. (In Chinese)

    MathSciNet  Google Scholar 

  32. X C Xiao, Y C Zhu, M L Ding. Erasures and equalities for fusion frames in Hilbert spaces, Bull Malays Math Sci Soc, 2015, 38(3): 1035–1045.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Shantou University, Shantou, 515063, China

    Xue-bin Li & Shou-zhi Yang

Authors
  1. Xue-bin Li
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  2. Shou-zhi Yang
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Correspondence to Shou-zhi Yang.

Additional information

Supported by the National Natural Science Foundation of China (11071152), the Natural Science Foundation of Guangdong Province (S2015A030313443).

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Li, Xb., Yang, Sz. Fusion-Riesz frame in Hilbert space. Appl. Math. J. Chin. Univ. 32, 339–352 (2017). https://doi.org/10.1007/s11766-017-3448-x

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  • DOI: https://doi.org/10.1007/s11766-017-3448-x

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