Continuous K-g-Frames in Hilbert Spaces

  • E. Alizadeh
  • A. RahimiEmail author
  • E. Osgooei
  • M. Rahmani
Original Paper


In this paper, we intend to introduce the concept of c-K-g-frames, which are the generalization of K-g-frames. In addition, we prove some new results on c-K-g-frames in Hilbert spaces. Moreover, we define the related operators of c-K-g-frames. Then, we give necessary and sufficient conditions on c-K-g-frames to characterize them. Finally, we verify the perturbation of c-K-g-frames.


c-K-g-frame K-g-frame g-Frame K-frame 

Mathematics Subject Classification

42C15 42C40 



The authors would like to thank Professor M. H. Faroughi for his comments and suggestions. We express also our special thanks to the reviewers due to their helpful comments for improving paper.


  1. 1.
    Abdollahpour, M.R., Faroughi, M.H.: Continuous G-frames in Hilbert spaces. Southeast Asian Bull. Math. 32, 1–19 (2008)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Ali, S.T., Antoine, J.P., Gazeau, J.P.: Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Asgari, M.S., Rahimi, H.: Generalized frames for operators in Hilbert spaces. Inf. Dimens. Anal. Quant. Probab. Relat. Top. 17(2), 1450013 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Balazs, P., Bayer, D., Rahimi, A.: Multipliers for countinuous frames in Hilbert spaces. J. Phys. A Math. Theor. 45, 2240023(20p) (2012)CrossRefGoogle Scholar
  5. 5.
    Casazza, P.G., Christensen, O.: Perturbation of operators and application to frame theory. J. Fourier Anal. Appl. 3, 543–557 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Douglas, R.G.: No majorization, factorization and range inclusion of operators on Hilbert space. Proc. Am. Math. Soc. 17(2), 413–415 (1966)zbMATHCrossRefGoogle Scholar
  7. 7.
    Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Fornasier, M., Rauhut, H.: Continuous frames, function spaces, and the discretization problem. J. Fourier Anal. Appl. 11(3), 245–287 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Gabardo, J.P., Han, D.: Frames associated with measurable space. Adv. Comput. Math. 18(3), 127–147 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gavruta, L.: Frames for operators. Appl. Comput. Harmon. Anal. 32, 139–144 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hua, D., Hung, Y.: K-g-frames and stability of K-g-frames in Hilbert spaces. J. Korean Math. Soc. 53(6), 1331–1345 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Kaiser, G.: A Friendly Guide to Wavelets. Birkhauser, Boston (1994)zbMATHGoogle Scholar
  13. 13.
    Li, D.-F., Yang, L.-J., Wu, G.: The unified condition for stability of g-frames. In: 2010 3rd International Conference on Advanced Computer Theory and Engineering (ICACTE), 2010 , Vol. 4, pp. V4-305–V4-308Google Scholar
  14. 14.
    Najati, A., Faroughi, M.H., Rahimi, A.: G-frames and stability of g-frames in Hilbert spaces. Methods. Funct. Anal. Topol. 14(3), 271–286 (2008)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Rahmani, M.: On some properties of c-frames. J. Math. Res. Appl. 37(4), 466–476 (2017)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rahmani, M.: Sum of c-frames, c-Riesz Bases and orthonormal mappings. U. P. B. Sci. Bull Ser. A 77(3), 3–14 (2015)Google Scholar
  17. 17.
    Rahimi, A., Najati, A., Dehghan, Y.N.: Continuous frames in Hilbert spaces. Methods. Funct. Anal. Topol. 12(2), 170–182 (2006)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Strohmer, T., Heath Jr., R.: Grassmannian frames with applications to conding and communications. Apple. Comput. Harmon. Anal. 14, 257–275 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Sun, W.C.: G-frames and g-Riesz. J. Math. Anal. Appl. 322(1), 437–452 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Xiao, X., Zhu, Y.: Exact K-g-frames in Hilbert spaces. Results Math. 72, 1329–1339 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Sun, W.: Stability of g-frame. J. Math. Anal. Appl. 326(2), 858–868 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Zhou, Y., Zhu, Y.: K-g-frames and dual g-frames for closed subspaces. Acta Math. Sin. Chin. Ser. 56(5), 799–806 (2013)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Zhou, Y., Zhu, Y.: Characterization of K-g-frames in Hilbert spaces. Acta Math. Sin. Chin. Ser. 57(5), 1031–1040 (2014)MathSciNetzbMATHGoogle Scholar

Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Shabestar BranchIslamic Azad UniversityShabestarIran
  2. 2.Department of MathematicsUniversity of MaraghehMaraghehIran
  3. 3.Faculty of ScienceUrmia University of TechnologyUrmiaIran
  4. 4.Young Researchers and Elite Club, Ilkhchi BranchIslamic Azad UniversityIlkhchiIran

Personalised recommendations