O-Cross Gram matrices with respect to \(\varvec{g}\)-frames

Abstract

The matrix representation of operators in Hilbert spaces is a useful tool in applications. It is important to present the matrix representation by sequences other than orthonormal bases. In this paper, we extend the matrix representation of operators using g-frames and investigate their invertibility and stability.

References

  1. 1.

    Abdollahi, A.; Rahimi, A.: Some results on \(g\)-frames in Hilbert spaces. Turk. J. Math. 34, 695–704 (2010)

    MATH  MathSciNet  Google Scholar 

  2. 2.

    Ali, S.T.; Antoine, J.-P.; Gazeau, J.-P.: Continous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)

    Article  Google Scholar 

  3. 3.

    Ali, S.T.; Antoine, J.-P.; Gazeau, J.-P.: Coherent States. Wavelets and Their Generalization. Graduate Texts in Contemporary Physics. Springer, New York (2000)

    Google Scholar 

  4. 4.

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

    MATH  MathSciNet  Google Scholar 

  5. 5.

    Arefijamaal, A.; Zekaee, E.: Image processing by alternate dual Gabor frames. Bull. Iran. Math. Soc. 42(6), 1305–1314 (2016)

    MATH  MathSciNet  Google Scholar 

  6. 6.

    Arefijamaal, A.; Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Balazs, P.: Matrix-representation of operators using frames. Sampl. Theory Signal Image Process. (STSIP) 7(1), 39–54 (2008)

    MATH  MathSciNet  Google Scholar 

  8. 8.

    Balazs, P.; Kreuzer, W.; Waubke, H.: A stochastic 2d-model for calculating vibrations in liquids and soils. J. Comput. Acoust. 15(3), 271–283 (2007)

    MATH  Article  MathSciNet  Google Scholar 

  9. 9.

    Benedetto, J.; Powell, A.; Yilmaz, O.: Sigm-delta quantization and finite frames. IEEE Trans. Inf. Theory. 52, 1990–2005 (2006)

    MATH  Article  Google Scholar 

  10. 10.

    Bodmannand, B.G.; Paulsen, V.I.: Frames, graphs and erasures. Linear Algebra Appl. 404, 118–146 (2005)

    MATH  Article  MathSciNet  Google Scholar 

  11. 11.

    Bolcskel, H.; Hlawatsch, F.; Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256–3268 (1995)

    Article  Google Scholar 

  12. 12.

    Candes, E.J.; Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities. Commun. Pure Appl. Anal. 56, 216–266 (2004)

    MATH  Google Scholar 

  13. 13.

    Casazza, P.G.; Kutyniok, G.; Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 132–144 (2008)

    MATH  Article  MathSciNet  Google Scholar 

  14. 14.

    Christensen, O.: Frames and Bases: An Introductory Course. Birkhäuser, Boston (2008)

    Google Scholar 

  15. 15.

    Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl. 195(2), 401–414 (1995)

    MATH  Article  MathSciNet  Google Scholar 

  16. 16.

    Conway, J.B.: A Course in Functional Analysis, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1990)

    Google Scholar 

  17. 17.

    Daubechies, I.: The wavelet transform, time frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)

    MATH  Article  MathSciNet  Google Scholar 

  18. 18.

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

    MATH  Article  MathSciNet  Google Scholar 

  19. 19.

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

    MATH  Article  MathSciNet  Google Scholar 

  20. 20.

    Feichtinger, H.G.; Grochenig, K.: Irregular sampling theorems and series expansion of band-limited functions. Math. Anal. Appl. 167, 530–556 (1992)

    MATH  Article  MathSciNet  Google Scholar 

  21. 21.

    Feichtinger, H.G.; Strohmer, T.: Gabor Analysis and Algorithms—Theory and Applications. Birkhauser, Boston (1998)

    Google Scholar 

  22. 22.

    Fillmore, P.A.; Williams, J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)

    MATH  Article  MathSciNet  Google Scholar 

  23. 23.

    Futamura, F.: Frame diagonalization of matrices. Linear Algebra Appl. 436(9), 3201–3214 (2012)

    MATH  Article  MathSciNet  Google Scholar 

  24. 24.

    Gohberg, I.; Goldberg, S.; Kaashoek, M.: Basic Classes of Linear Operators. Birkhäuser, Basel (2003)

    Google Scholar 

  25. 25.

    Jin, G.; Chen, A.: Some basic properties of block operator matrices. arXiv:1403.7732

  26. 26.

    Kirkup, S.M.; Wadsworth, M.: Solution of inverse diffusion problems by operator-splitting methods. Appl. Math. Model. 26, 1003–1018 (2002)

    MATH  Article  Google Scholar 

  27. 27.

    Mitchell, A.R.; Griffiths, D.F.: The Finite Difference Method in Partial Differential Equations. Wiley, New York (1980)

    Google Scholar 

  28. 28.

    Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, Cambridge (1999)

    Google Scholar 

  29. 29.

    Najati, A.; Faroughi, M.H.; Rahimi, A.: \(g\)-frames and stability of \(g\)-frames in Hilbert spaces. Methods Funct. Anal. Topol. 4(3), 271–286 (2008)

    MATH  MathSciNet  Google Scholar 

  30. 30.

    Radol, K.: Matrices related to some fock space operators. Opuscula Math. 2, 289–297 (2011)

    Article  MathSciNet  Google Scholar 

  31. 31.

    Shamsabadi, M., Arefijamaal, A., Balazs, P., Rahimi, A.: \(U\)-cross Gram matrices and their associated reconstructions. arXiv:1804.00203

  32. 32.

    Sowa, A.: Encoding spatial data into quantum observables. arXiv:1609.01712

  33. 33.

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

    MATH  Article  MathSciNet  Google Scholar 

  34. 34.

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

    MATH  Article  MathSciNet  Google Scholar 

  35. 35.

    Zhou, P.: Numerical Analysis of Electromagnetic Fields. Springer, New York (1993)

    Google Scholar 

  36. 36.

    Zhu, Y.C.: Characterizations of \(g\)-frames and \(g\)-bases in Hilbert spaces. Acta Math. Sin. 24, 1727–1736 (2008)

    MATH  Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors are thankful to P. Balazs for his valuable comments. They also want to thank all anonymous reviewers concerned with this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ali Akbar Arefijamaal.

Additional information

Publisher's Note

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

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Arefijamaal, A.A., Shamsabadi, M. O-Cross Gram matrices with respect to \(\varvec{g}\)-frames. Arab. J. Math. 9, 259–269 (2020). https://doi.org/10.1007/s40065-019-0246-8

Download citation

Mathematics Subject Classification

  • 41A58
  • 43A35