Skip to main content
SpringerLink
Log in

Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces

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

Abstract

Weaving Hilbert space frames have been introduced recently by Bemrose et al. to deal with some problems in distributed signal processing. In this paper, we survey this topic from the viewpoint of the duality principle. In this regard, not only we obtain new properties in weaving frame theory related to dual frames but also we bring up new approaches for manufacturing pairs of woven frames. Specifically, we give some sufficient conditions under which a frame with its canonical dual, alternate duals or approximate duals constitute some concrete pairs of woven frames. Moreover, we provide some approaches for constructing weaving frames by using small perturbations and present a condition where different operators preserve the weaving property. As a consequence, the canonical duals of two woven frames are woven.

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

    Article  MathSciNet  Google Scholar 

  2. Arabyani-Neyshaburi, F., Arefijamaal, A.: Some constructions of \(K\)-frames and their duals. Rocky Mt. J. Math. 47(6), 1749–1764 (2017)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Arefijamaal, A., Arabyani-Neyshaburi, F.: Some properties of dual and approximate dual of fusion frames. Turk. J. Math. 41, 1191–1203 (2017)

    Article  MathSciNet  Google Scholar 

  5. Balan, R., Casazza, P.G., Heil, C., Landau, Z.: Deficits and excesses of frames. Adv. Comput. Math. 18, 93–116 (2003). (Special issue on frames)

    Article  MathSciNet  Google Scholar 

  6. Bemrose, T., Casazza, P.G., Grochenig, K., Lammers, M.C., Lynch, R.G.: Weaving Hilbert space frames. Oper. Matrices 10(4), 1093–1116 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  8. Bodmann, B.G., Casazza, P.G.: The road to equal-norm Parseval frames. J. Funct. Anal. 258(2), 397–420 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Bolcskel, H., Hlawatsch, F., Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256–3268 (1998)

    Article  Google Scholar 

  11. Cahill, J., Casazza, P.G., Kutyniok, G.: Operators and frames. J. Oper. Theory 70(1), 145–164 (2013)

    Article  MathSciNet  Google Scholar 

  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. Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–114 (2004)

    Article  MathSciNet  Google Scholar 

  14. Casazza, P.G., Lynch, R.G.: Weaving properties of Hilbert space frames In: Proceedings of the SampTA, pp. 110–114 (2015)

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

    Book  Google Scholar 

  16. Christensen, O., Laugesen, R.S.: Approximately dual frames in Hilbert spaces and applications to Gabor frames. Sampl. Theory Signal Image Process. 9, 77–89 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Ghaani Farashahi, A.: Square-integrability of multivariate metaplectic wave-packet representations. J. Phys. A Math. Theor. 50(115202), 1–22 (2017)

    MathSciNet  MATH  Google Scholar 

  18. Ghaani Farashahi, A.: Square-integrability of metaplectic wave-packet representations on \(L^{2}(R)\). J. Math. Anal. Appl. 449(1), 769–79 (2017)

    Article  MathSciNet  Google Scholar 

  19. Ghaani Farashahi, A.: Multivariate wave-packet transforms. J. Anal. Appl. 36(4), 481–500 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Găvruţa, L.: Frames for operators. Appl. Comput. Harmon. Anal. 32, 139–144 (2012)

    Article  MathSciNet  Google Scholar 

  21. Javanshiri, H.: Some properties of approximate dual frames in Hilbert spaces. Results Math. 70, 475–485 (2016)

    Article  MathSciNet  Google Scholar 

  22. Kaftal, V., Larson, D.R., Zhang, Sh: Operator-valued frames. Trans. Am. Math. Soc. 361, 6349–6385 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Vashisht, L.K., Deepshikha: On weaving frames. Houston J. Math. (to appear)

  25. Vashisht, L.K., Deepshikha, Garg, S., Daus, P.K.: On genralized weaving frames of Hilbert spaces. Rocky Mt. J. Math. (to appear)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

    Fahimeh Arabyani-Neyshaburi & Ali Akbar Arefijamaal

Authors
  1. Fahimeh Arabyani-Neyshaburi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ali Akbar Arefijamaal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ali Akbar Arefijamaal.

Additional information

Communicated by Sorina Barza.

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

Arabyani-Neyshaburi, F., Arefijamaal, A.A. Manufacturing Pairs of Woven Frames Applying Duality Principle on Hilbert Spaces. Bull. Malays. Math. Sci. Soc. 44, 147–161 (2021). https://doi.org/10.1007/s40840-020-00940-9

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40840-020-00940-9

Keywords

Mathematics Subject Classification

Search

Navigation