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On Some New Inequalities For Continuous Fusion Frames in Hilbert Spaces

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Abstract

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.

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References

  1. Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert space. Ann. Phys. 222(1), 1–37 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antoine, J.P., Balazs, P.: Frames and semi-frames. J. Phys. A 44(20), 205201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Balan, R., Casazza, P., Edidin, D.: On signal reconstruction without phase. App. Comput. Harmon. Anal. 20(3), 345–356 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  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. Balan, R., Casazza, P., Edidin, D., Kutyniok, G.: A new identity for Parseval frames. Proc. Am. Math. Soc. 135(4), 1007–1015 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bölcskei, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46(12), 3256–3268 (1998)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Candès, E.J.: Harmonic analysis of neural networks. Appl. Comput. Harmon. Anal. 6(2), 197–218 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–114 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Casazza, P.G., Kutyniok, G., Li, S.-D.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 114–132 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daubechies, I., Grossmann, A., Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27(5), 1271–1283 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72(2), 341–366 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  14. Elron, N., Eldar, Y.C.: Optimal encoding of classical information in a quantum medium. IEEE Trans. Inf. Theory 53(5), 1900–1907 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Faroughi, M.H., Reza, A.: C-fusion frame. J. Appl. Sci. 8(16), 2881–2887 (2008)

    Article  Google Scholar 

  16. Gabardo, J.P., Han, D.-G.: Frames associated with measurable spaces. Adv. Comput. Math. 18(2–4), 127–147 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Găvruţa, P.: On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl. 321(1), 469–478 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guo, Q.-P., Leng, J.-S., Li, H.-B.: Some equalities and inequalities for fusion frames. SpringerPlus 5(1), 121 (2016)

    Article  Google Scholar 

  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)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Poria, A.: Some identities and inequalities for Hilbert Schmidt frames. Mediterr. J. Math. 14(2), 59 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  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)

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Xiang, Z.-Q.: New inequalities for g-frames in Hilbert C*-modules. J. Math. Inequal. 10(3), 889–897 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  27. Zhang, W., Li, Y.-Z.: Some new inequalities for continuous fusion frames and fusion pairs. SpringerPlus 5(1), 1600 (2016)

    Article  Google Scholar 

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Acknowledgements

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

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Authors and Affiliations

  1. School of Mathematics, HeFei University of Technology, No. 193, Tunxi Road, Hefei, 230009, Anhui, People’s Republic of China

    Dongwei Li

  2. School of Mathematical Sciences, University of Electronic Science and Technology of China, No.2006, Xiyuan Ave, West Hi-Tech Zone, Chengdu, 611731, Sichuan, People’s Republic of China

    Dongwei Li & Jinsong Leng

Authors
  1. Dongwei Li
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  2. Jinsong Leng
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Correspondence to Dongwei Li.

Additional information

This work was supported by the National Natural Science Foundation of China (11271001) and the National Natural Science Foundation of China (61370147).

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Li, D., Leng, J. On Some New Inequalities For Continuous Fusion Frames in Hilbert Spaces. Mediterr. J. Math. 15, 173 (2018). https://doi.org/10.1007/s00009-018-1219-4

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  • DOI: https://doi.org/10.1007/s00009-018-1219-4

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