Skip to main content
SpringerLink
Log in

On the Sum of K-Frames in Hilbert Spaces

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In recent years, research on a special class of frames, named K-frames where K is an operator, has become significant in theory and applications. Since the finite sum of K-frames may not be a K-frame for the Hilbert space, in this paper, we discuss the sum and stability of K-frames in Hilbert spaces. First, we obtain some sufficient conditions for the finite sum of a K-frame and a Bessel sequence to be a K-frame. Then we get the K-dual of the sum of K-frames by the dual of the original K-frames. In particular, we give some new results about the operator K and the analysis operator in the discussion. Moreover, we discuss the stability of K-frames and get some conclusions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

  2. Holmes, R.B., Paulsen, V.I.: Optimal frames for erasures. Linear Algebra Appl. 377(1), 31–51 (2004)

    Article  MathSciNet  Google Scholar 

  3. Li, D., Leng, J., Huang, T.: Frame expansions with probabilistic erasures. Digit. Signal Process. 72, 75–82 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bodmann, B.G., Paulsen, V.I.: Frame paths and error bounds for sigma-delta quantization. Appl. Comput. Harmon. Anal. 22(2), 176–197 (2007)

    Article  MathSciNet  Google Scholar 

  5. Dana, A.F., Gowaikar, R., Palanki, R.: Capacity of wireless erasure networks. IEEE Trans. Inf. Theory 52(3), 789–804 (2006)

    Article  MathSciNet  Google Scholar 

  6. Leng, J., Han, D., Huang, T.: Probability modelled optimal frames for erasures. Linear Algebra Appl. 438(11), 4222–4236 (2013)

    Article  MathSciNet  Google Scholar 

  7. Leng, J., Huang, T.: Construction of fusion frame systems in finite dimensional Hilbert spaces. Abstract Appl. Anal. 2014, 1–12 (2014)

    Article  MathSciNet  Google Scholar 

  8. Li, D., Jingsong, L., Huang, T.: Some equalities and inequalities for probabilistic frames. J. Inequal. Appl. 2016(1), 245 (2016)

    Article  MathSciNet  Google Scholar 

  9. Leng, J., Han, D., Huang, T.: Optimal dual frames for communication coding with probabilistic erasures. IEEE Trans. Signal Process. 59(11), 5380–5389 (2011)

    Article  MathSciNet  Google Scholar 

  10. Fornasier, M.: Quasi-orthogonal decompositions of structured frames. Math. Anal. Appl. 289(1), 180–199 (2004)

    Article  MathSciNet  Google Scholar 

  11. Upenderreddy, G., Gopalreddy, N., Krishnareddy, B.: Frame operator and Hilbert–Schmidt operator in tensor product of Hilbert spaces. J. Dyn. Syst. Geom. Theor. 7(1), 61–70 (2009)

    MathSciNet  Google Scholar 

  12. Khosravi, A., Moslemipour, N.A.: Frame operator and alternate dual modular frame. Int. J. Appl. Math. 13(2), 177–189 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Li, X.B., Yang, S.Z., Zhu, Y.C.: Some results about operator perturbation of fusion frames in Hilbert spaces. J. Math. Anal. Appl. 421(2), 1417–1427 (2015)

    Article  MathSciNet  Google Scholar 

  14. Arefijamaal, A.A., Ghasemi, S.: On characterization and stability of alternate dual of g-frames. Turk. J. Math. 37(1), 71–79 (2013)

    MathSciNet  MATH  Google Scholar 

  15. Xiao, X.C., Ding, M.L., Zhu, Y.C.: Several studies of continuous K-frame in Hilbert spaces. J. Anhui Univ. (2013)

  16. Gavruta L.: New results on frames for operators. In: Proc Intern Conf Sciences, vol. 2 (2012)

  17. Guo, X.X.: Canonical dual K Bessel sequences and dual K Bessel generators for unitary systems of Hilbert spaces. J. Math. Anal. Appl. 444(1), 598–609 (2016)

    Article  MathSciNet  Google Scholar 

  18. Liang, L.: The optimal K-duals for 1-erasure for parseval K-frames. Adv. Appl. Math. 03(4), 192–200 (2014)

    Article  Google Scholar 

  19. Xiao, X.C., Zhu, Y.C., Găvruţa, L.: Some properties of K-frames in Hilbert spaces. Results Math. 63, 1243–1255 (2013)

    Article  MathSciNet  Google Scholar 

  20. Liang, L., Pengtong, L.: K-frames and K-duals in Hilbert spaces. J. Nanjing Univ. 32(1), 77–91 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Li, D., Leng, J., Huang, T.: On sum and stability of g-frames in Hilbert spaces. Linear Multilinear Algebra 66(8), 1578–1592 (2018)

    Article  MathSciNet  Google Scholar 

  22. Sun, W.: Stability of g-frames. Math. Anal. Appl. 326(2), 858–868 (2007)

    Article  MathSciNet  Google Scholar 

  23. Xi, Y.Y., Mei, Q.H.: The sums of K-frame vector for unitary system in Hilbert spaces. J. Yuncheng Univ. 35(6), 52–56 (2017)

    Google Scholar 

Download references

Funding

This research is supported by National Natural Science Foundation of China (LJT10110010115).

Author information

Authors and Affiliations

  1. University of Electronic Science and Technology of China, Chengdu, China

    Miao He, Jinsong Leng, Jiali Yu & Yuxiang Xu

Authors
  1. Miao He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jinsong Leng
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiali Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuxiang Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jinsong Leng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, M., Leng, J., Yu, J. et al. On the Sum of K-Frames in Hilbert Spaces. Mediterr. J. Math. 17, 46 (2020). https://doi.org/10.1007/s00009-020-1487-7

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-020-1487-7

Keywords

Mathematics Subject Classification

Search

Navigation