On Some New Inequalities For Continuous Fusion Frames in Hilbert Spaces

  • Dongwei LiEmail author
  • Jinsong Leng


Continuous frames and fusion frames were considered recently as generalizations of frames in Hilbert spaces. In this paper, for any continuous fusion frame, we obtain a new family of inequalities which are parametrized by a parameter \(\lambda \in \mathbb {R}\). By suitable choices of \(\lambda \), one obtains the previous results as special cases. Moreover, these new inequalities involve the expressions \(\langle S_Yh,h \rangle \), \(\Vert S_Yh\Vert \), etc., where \(S_Y\) is a “truncated form” of the continuous fusion frame operator.


Continuous fusion frames fusion frames dual frames operators 

Mathematics Subject Classification

Primary 42C15 Secondary 46C07 



The authors would like to thank the reviewer for several helpful suggestions that helped improve the presentation of this paper.


  1. 1.
    Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert space. Ann. Phys. 222(1), 1–37 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antoine, J.P., Balazs, P.: Frames and semi-frames. J. Phys. A 44(20), 205201 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. App. Comput. Harmon. Anal. 20(3), 345–356 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balan, R., Casazza, P., Edidin, D., Kutyniok, G.: Decompositions of frames and a new frame identity. Proc. SPIE 5914, 1–10 (2005)Google Scholar
  5. 5.
    Balan, R., Casazza, P., Edidin, D., Kutyniok, G.: A new identity for Parseval frames. Proc. Am. Math. Soc. 135(4), 1007–1015 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Balazs, P., Antoine, J.P., Gryboś, A.: Weighted and controlled frames: Mutual relationship and first numerical properties. Int. J. Wavelets Multiresolut. Inf. Proc. 8(01), 109–132 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46(12), 3256–3268 (1998)CrossRefGoogle Scholar
  8. 8.
    Boufounos, P., Kutyniok, G., Rauhut, H.: Sparse recovery from combined fusion frame measurements. IEEE Trans. Inf. Theory 57(6), 3864–3876 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Candès, E.J.: Harmonic analysis of neural networks. Appl. Comput. Harmon. Anal. 6(2), 197–218 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–114 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Casazza, P.G., Kutyniok, G., Li, S.-D.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 114–132 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72(2), 341–366 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Elron, N., Eldar, Y.C.: Optimal encoding of classical information in a quantum medium. IEEE Trans. Inf. Theory 53(5), 1900–1907 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Faroughi, M.H., Reza, A.: C-fusion frame. J. Appl. Sci. 8(16), 2881–2887 (2008)CrossRefGoogle Scholar
  16. 16.
    Gabardo, J.P., Han, D.-G.: Frames associated with measurable spaces. Adv. Comput. Math. 18(2–4), 127–147 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Găvruţa, P.: On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl. 321(1), 469–478 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guo, Q.-P., Leng, J.-S., Li, H.-B.: Some equalities and inequalities for fusion frames. SpringerPlus 5(1), 121 (2016)CrossRefGoogle Scholar
  19. 19.
    Leng, J.-S., Guo, Q.-X., Huang, T.-Z.: The duals of fusion frames for experimental data transmission coding of high energy physics. Adv. High Energy Phys. 2013(5), 178–182 (2013)zbMATHGoogle Scholar
  20. 20.
    Leng, J.-S., Han, D.-G., Huang, T.-Z.: Optimal dual frames for communication coding with probabilistic erasures. IEEE Trans. Signal Process. 59(11), 5380–5389 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Li, D.-W., Leng, J.-S., Huang, T.-Z., Xu, Y.-X.: Some equalities and inequalities for probabilistic frames. J. Inequal. Appl. 2016(1), 1–11 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Poria, A.: Some identities and inequalities for Hilbert Schmidt frames. Mediterr. J. Math. 14(2), 59 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rozell, C.J., Johnson, D. H.: Analysis of noise reduction in redundant expansions under distributed processing requirements. In: Proceedings (ICASSP’05) IEEE international conference on acoustics, speech, and signal processing, 2005., vol 4, pp. iv–185, IEEE (2005)Google Scholar
  24. 24.
    Sun, W.-C.: G-frames and g-Riesz bases. J. Math. Anal. Appl. 322(1), 437–452 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xiang, Z.-Q.: New inequalities for g-frames in Hilbert C*-modules. J. Math. Inequal. 10(3), 889–897 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yang, X.-H., Li, D.-F.: Some new equalities and inequalities for g-frames and their dual frames. Acta Math. Sinica Chin. Ser. 52(5), 1033–1040 (2009)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Zhang, W., Li, Y.-Z.: Some new inequalities for continuous fusion frames and fusion pairs. SpringerPlus 5(1), 1600 (2016)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of MathematicsHeFei University of TechnologyHefeiPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of Electronic Science and Technology of ChinaChengduPeople’s Republic of China

Personalised recommendations