Skip to main content
SpringerLink
Log in

On R-duals of Type III in Hilbert Spaces

  • Research Articles
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

Following work by Casazza, Kutyniok, and Lammers and its development by Stoeva and Christensen, we provide some novel characterizations of R-dual sequences of type III in Hilbert spaces. We systematically extend the construction procedure by basing it on a choice of an antiunitary involution. For certain classes of R-duals of type III, we derive a representation of the associated frame operator in terms of spectral measures.

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. P. G. Casazza, G. Kutyniok, and M. Lammers, “Duality principles in frame theory”, J. Fourier Anal. Appl., 10 (2004), 383–408.

    Article  MathSciNet  Google Scholar 

  2. P. G. Casazza, G. Kutyniok, and M. Lammers, “Duality principles, localization of frames, and Gabor theory”, SPIE Proceedings in Wavelets XI, 5914 2005.

    Google Scholar 

  3. P. G. Casazza, “Modern tools for Weyl-Heisenberg (Gabor) frame theory”, Adv. in Imag. and Electron. Physics, 115 (2001), 1–127.

    Article  Google Scholar 

  4. O. Christensen, Frame and Bases, An Introductory Course, Birkhäuser, Basel, 2008.

    MATH  Google Scholar 

  5. O. Christensen, H. O. Kim, and R. Y. Kim, “On the duality principle by Casazza, Kutyniok, and Lammers”, J. Fourier Anal. Appl., 17 (2011), 640–655.

    Article  MathSciNet  Google Scholar 

  6. O. Christensen, X. C. Xiao, and Y. C. Zhu, “Characterizing R-duality in Banach spaces”, Acta Math. Sin. Engl. Ser., 1 (2013), 75–84.

    Article  MathSciNet  Google Scholar 

  7. Z. Chuang and J. Zhao, “On equivalent conditions of two sequences to be R-dual”, J. Inequal. Appl., 10 (2015), 1–8.

    MathSciNet  MATH  Google Scholar 

  8. D. T. Stoeva and O. Christensen, “On R-duals and the Duality principle in Gabor Analysis”, J. Fourier Anal. Appl., 21 (2015), 383–400.

    Article  MathSciNet  Google Scholar 

  9. E. Kreyszig, Introductory Functionl Analysis with Approximation, John Wiley, New York, 1978.

    Google Scholar 

Download references

Acknowledgments

The authors would like to thank the referees for valuable comments and suggestions for improvement of the manuscript.

Author information

Authors and Affiliations

  1. Lehrstuhl A für Mathematik, RWTH Aachen University, Aachen, Germany, Aachen, Germany

    H. Führ

  2. Department of Mathematics, Payame Noor University, Tehran, Iran, Tehran, Iran

    J. Cheshmavar & A. Akbarnia

Authors
  1. H. Führ
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Cheshmavar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Akbarnia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to H. Führ, J. Cheshmavar or A. Akbarnia.

Additional information

Translated from Funktsional'nyi Analiz i ego Prilozheniya, 2021, Vol. 55, pp. 62–74 https://doi.org/10.4213/faa3835.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Führ, H., Cheshmavar, J. & Akbarnia, A. On R-duals of Type III in Hilbert Spaces. Funct Anal Its Appl 55, 226–235 (2021). https://doi.org/10.1134/S0016266321030059

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0016266321030059

Keywords

Search

Navigation