Korean Journal of Mathematics

  • 제30권1호
  • /
  • Pages.91-107
  • /
  • 2022
  • /
  • 1976-8605(pISSN)
  • /
  • 2288-1433(eISSN)
강원경기수학회 (The Kangwon-Kyungki Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

CONTROLLED K-FRAMES IN HILBERT C*-MODULES

  • Rajput, Ekta (Dhirubhai Ambani Institute of Information and Communication Technology) ;
  • Sahu, Nabin Kumar (Dhirubhai Ambani Institute of Information and Communication Technology) ;
  • Mishra, Vishnu Narayan (Department of Mathematics, Indira Gandhi National Tribal University)
  • 투고 : 2021.10.04
  • 심사 : 2022.02.08
  • 발행 : 2022.03.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Controlled frames have been the subject of interest because of their ability to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we introduce the notion of controlled K-frame or controlled operator frame in Hilbert C*-modules. We establish the equivalent condition for controlled K-frame. We investigate some operator theoretic characterizations of controlled K-frames and controlled Bessel sequences. Moreover, we establish the relationship between the K-frames and controlled K-frames. We also investigate the invariance of a controlled K-frame under a suitable map T. At the end, we prove a perturbation result for controlled K-frame.

키워드

과제정보

The authors are thankful to the anonymous referees for their valuable suggestions and comments that significantly improved the presentation and correctness of this paper.

참고문헌

  1. A. Najati, M. M. Saem and P. Gavruta, Frames and operators in Hilbert C*-modules, Operators and Matrices 10 (1) (2016), 73-81.
  2. D. Han, W. Jing and R. Mohapatra, Perturbation of frames and Riesz bases in Hilbert C*-modules, Linear Algebra Appl. 431 (2009), 746-759. https://doi.org/10.1016/j.laa.2009.03.025
  3. D. Han, W. Jing, D. Larson and R. Mohapatra, Riesz bases and their dual modular frames in Hilbert C*-modules, J. Math Anal. Appl. 343 (2008), 246-256. https://doi.org/10.1016/j.jmaa.2008.01.013
  4. E. J. Candes and D. L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Commun. Pure Appl. Math. 57 (2) (2004), 219-266. https://doi.org/10.1002/cpa.10116
  5. H. Bolcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46 (12) (1998), 3256-3268. https://doi.org/10.1109/78.735301
  6. I. Daubechies, A. Grossmann and Y. Meyer, Painless non orthogonal expansions, J. Math.Phys. 27 (1986), 1271-1283. https://doi.org/10.1063/1.527388
  7. I. Kaplansky, Algebra of type I, Ann. Math. 56 (1952), 460-472. https://doi.org/10.2307/1969654
  8. L. Arambaic, On frames for countably generated Hilbert C*-modules, Proc. Amer. Math Soc. 135 (2007), 469-478. https://doi.org/10.1090/S0002-9939-06-08498-X
  9. L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012), 139-144. https://doi.org/10.1016/j.acha.2011.07.006
  10. M. Frank and D. R. Larson, Frames in Hilbert C*-modules and C*-algebras, J. Operator Theory 48 (2002), 273-314.
  11. M. Nouri, A. Rahimi and Sh. Najafzadeh, Controlled K-frames in Hilbert Spaces, J. of Ramanujan Society of Math. and Math. Sc. 4 (2) (2015), 39-50.
  12. M. R. Kouchi and A. Rahimi, On controlled frames in Hilbert C*-modules, Int. J. Walvelets Multi. Inf. Process. 15 (4) (2017), 1750038. https://doi.org/10.1142/S0219691317500382
  13. P. Balazs, J-P. Antoine and A. Grybos, Weighted and Controlled Frames, Int. J. Walvelets Multi. Inf. Process. 8 (1) (2010), 109-132. https://doi.org/10.1142/S0219691310003377
  14. P. J. S. G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In: Signals Processing for Multimedia J. S. Byrnes(Ed.)(1999),35-54.
  15. R. J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Math.Soc. 72 (1952), 341-366. https://doi.org/10.1090/S0002-9947-1952-0047179-6
  16. T. Strohmer and R. Jr. Heath, Grassmanian frames with applications to coding and communications, Appl. Comput. Harmon. Anal. 14 (2003), 257-275. https://doi.org/10.1016/S1063-5203(03)00023-X
  17. W. Jing, Frames in Hilbert C*-modules, Ph.D. Thesis, University of Central Frorida, (2006).
  18. W. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. https://doi.org/10.1090/S0002-9947-1973-0355613-0
  19. X. Fang, J. Yu and H. Yao, Solutions to operator equations on Hilbert C-modules, Linear Alg. Appl. 431 (11) (2009), 2142-2153. https://doi.org/10.1016/j.laa.2009.07.009
  20. Y. C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Walvelets Multi. Inf. Process. 3 (3) (2005), 347-359. https://doi.org/10.1142/S0219691305000890
  21. Y. C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier. Anal. Appl. 9 (1) (2003), 77-96. https://doi.org/10.1007/s00041-003-0004-2