Skip to main content
SpringerLink
Log in

Portraits and Perturbations of Hilbert–Schmidt Frame Sequences

  • Published:
Bulletin of the Malaysian Mathematical Sciences Society Aims and scope Submit manuscript

Abstract

It is well known that there are two approaches applicable in constructing frames starting from one fixed frame. One is based on \(l^{2}\)-operator portraits by which, using a suitable bounded linear operator on \(l^{2}\), one can construct an arbitrary frame from one fixed frame. The other is based on perturbation that allows suitable perturbing a frame leaving a frame. The study of Hilbert–Schmidt frames (HS-frames) has interested some mathematicians in recent years. This paper addresses \(l^{2}\)-operator portraits and perturbations in the setting of HS-frames. We prove that the portrait of a HS-frame under a bounded invertible operator on \(l^{2}\) is still a HS-frame; present a sufficient condition on bounded operators on \(l^{2}\) which transform an \(l^{2}\)-decomposable HS-frame into another HS-frame (HS-Riesz basis, HS-frame sequence and HS-Riesz sequence); and prove that suitable perturbing a HS-frame sequence (HS-Riesz sequence) leaves a HS-frame sequence (HS-Riesz sequence). Finally, using these results we recover some conclusions on frames.

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. Abdollahpour, M.R., Khedmati, Y.: g-duals of continuous g-frames and their perturbations. Results Math. 73(4), 15 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aldroubi, A.: Portraits of frames. Proc. Am. Math. Soc. 123(6), 1661–1668 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aldroubi, A., Cabrelli, C., Molter, U.: Wavelets on irregular grids with arbitrary dilation matrices and frame atoms for \(L^{2}({\mathbb{R} }^{d})\). Appl. Comput. Harmon. Anal. 17(2), 119–140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arefijamaal, A.A., Sadeghi, Gh.: von Neumann-Schatten dual frames and their perturbations. Results Math. 69(3–4), 431–441 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Asgari, M.S., Khosravi, A.: Frames and bases of subspaces in Hilbert spaces. J. Math. Anal. Appl. 308(2), 541–553 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cazassa, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. 3(5), 543–557 (1997)

    Article  MathSciNet  Google Scholar 

  7. Casazza, P.G., Kutyniok, G.: Frames of subspaces. Wavelets, frames and operator theory, Contemp. Math., Vol. 345, Amer. Math. Soc., Providence, RI, 87–113 (2004)

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Christensen, O.: A Paley-Wiener theorem for frames. Proc. Am. Math. Soc. 123(7), 2199–2201 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Christensen, O.: Frame perturbations. Proc. Am. Math. Soc. 123(4), 1217–1220 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Christensen, O.: Operators with closed range, pseudo-inverses, and perturbation of frames for a subspace. Canad. Math. Bull. 42(1), 37–45 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Christensen, O.: An Introduction to Frames and Riesz Bases. Second edition, Birkhäuser (2016)

  13. Christensen, O., Hasannasab, M.: Operator representations of frames: boundedness, duality, and stability. Integral Equ. Op. Theory 88(4), 483–499 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Christensen, O., Lennard, C., Lewis, C.: Perturbation of frames for a subspace of a Hilbert space. Rocky Mountain J. Math. 30(4), 1237–1249 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Dong, J., Li, Y.-Z.: Duality principles in Hilbert-Schmidt frame theory. Math. Methods Appl. Sci. 44(6), 4888–4906 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dörfler, M., Feichtinger, H.G., Gröchenig, K.: Time-frequency partitions for the Gelfand triple \((S_{0}, L^{2}, S^{\prime }_{0})\). Math. Scand. 98(1), 81–96 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Favier, S.J., Zalik, R.A.: On the stability of frames and Riesz bases. Appl. Comput. Harmon. Anal. 2(2), 160–173 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Guo, X.: Perturbations of invertible operators and stability of g-frames in Hilbert spaces. Results Math. 64(3–4), 405–421 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo, X.: Operator parameterizations of g-frames. Taiwanese J. Math. 18(1), 313–328 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Heil, C.: A Basis Theory Primer. Birkhäuser/Springer, New York (2011)

    Book  MATH  Google Scholar 

  24. Javanshiri, H., Hajiabootorabi, M., Mardanbeigi, M.R.: The effect of perturbations of frames on their alternate and approximately dual frames. Math. Methods Appl. Sci. 45(4), 2058–2071 (2022)

    Article  MathSciNet  Google Scholar 

  25. Jiang, Z., Yang, S.Z., Zheng, X.W.: Perturbation and construction of almost self-located robust frames with applications to erasure recovery. Math. Methods Appl. Sci. 44(8), 7333–7342 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kato, T.: Perturbation Theory for Linear Operator. Springer-Verlag, New York (1984)

    Google Scholar 

  27. Li, D.W., Leng, J.S., Huang, T.Z., Sun, G.M.: On sum and stability of g-frames in Hilbert spaces. Linear Multilinear Algebra 66(8), 1578–1592 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  28. Li, Y.-N., Li, Y.-Z.: Hilbert-Schmidt frames and their duals. Int. J. Wavelets Multiresolut. Inf. Process. 19(5), 2150011 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  29. Li, S., Ogawa, H.: Pseudoframes for subspaces with applications. J. Fourier Anal. Appl. 10(4), 409–431 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  30. Li, Y.-Z., Zhang, X.-L.: Frame properties of Hilbert-Schmidt operator sequences, Mediterr. J. Math., accepted

  31. Li, Y.-Z., Zhang, X.-L.: Dilations of (dual) Hilbert-Schmidt frames. Ann. Funct. Anal. 13(3), 20 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  32. Poria, A.: Approximation of the inverse frame operator and stability of Hilbert-Schmidt frames. Mediterr. J. Math. 14(4), 22 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. Sadeghi, Gh., Arefijamaal, A.A.: von Neumann-Schatten frames in separable Banach spaces. Mediterr. J. Math. 9(3), 525–535 (2012)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, Y.J., Zhu, Y.C.: G-frames and g-frame sequences in Hilbert spaces. Acta Math. Sin. 25(12), 2093–2106 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  38. Xiang, Z.Q.: Some new results on the construction and stability of K-g-frames in Hilbert spaces. Int. J. Wavelets Multiresolut. Inf. Process. 18(5), 2050034, 19 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  39. Young, R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)

    MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to extend their sincere gratitude to referees for their careful reading this article and valuable comments, which helped to greatly improve the readability of this article.

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Beijing University of Technology, Beijing, 100124, People’s Republic of China

    Xiao-Li Zhang & Yun-Zhang Li

Authors
  1. Xiao-Li Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yun-Zhang Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yun-Zhang Li.

Additional information

Communicated by Rosihan M. Ali.

Publisher's Note

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

Supported by National Natural Science Foundation of China (Grant No. 11971043)

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, XL., Li, YZ. Portraits and Perturbations of Hilbert–Schmidt Frame Sequences. Bull. Malays. Math. Sci. Soc. 45, 3197–3223 (2022). https://doi.org/10.1007/s40840-022-01375-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-022-01375-0

Keywords

Mathematics Subject Classification

Search

Navigation