Skip to main content
Springer Nature Link
Log in

Bessel Sequences and Frames in Semi-inner Product Spaces

  • Conference paper
  • First Online:
Mathematics and Computing (ICMC 2017)

Part of the book series: Communications in Computer and Information Science ((CCIS,volume 655))

Included in the following conference series:

  • 952 Accesses

Abstract

In this paper, the concept of Bessel sequence and frame are introduced in semi-inner product spaces. Some properties of the Bessel sequences and frame are investigated in smooth uniformly convex Banach spaces. One characterization of the space of all Bessel sequences has been pointed out. Examples of frames are constructed in the real sequence spaces \(l^{p}\), \(1<p<\infty \).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldroubi, A., Sun, Q., Tang, W.S.: \(p\)-frames and shift invariant spaces of \(L^{p}\). J. Fourier Anal. Appl. 7, 1–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)

    MATH  Google Scholar 

  3. Cao, H.X., Li, L., Chen, Q.J., Ji, G.X.: \((p, Y)\)-operator frames for a Banach space. J. Math. Anal. Appl. 347, 583–591 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Carando, D., Lassalle, S., Schmidberg, P.: The reconstruction formula for Banach frames and duality. J. Approx. Theory 163, 640–651 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Casazza, P.G., Christensen, O.: Frames containing a Riesz basis and preservation of this property under perturbation. SIAM J. Math. Anal. 29, 266–278 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Casazza, P.G., Christensen, O., Stoeva, D.T.: Frame expansions in separable Banach spaces. J. Math. Anal. Appl. 307(2), 710–723 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Christensen, O.: Frames perturbations. Proc. Amer. Math. Soc. 123, 1217–1220 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  11. Christensen, O.: A Paley-Wiener theorem for frames. Proc. Amer. Math. Soc. 123, 2199–2202 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nachr. 185, 33–47 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Fornasier, M.: Banach frames for \(\alpha \)-modulation spaces. Appl. Comput. Harmon. Anal. 22, 157–175 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. Giles, J.R.: Classes of semi-inner product spaces. Trans. Amer. Math. Soc. 129, 436–446 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Grochenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Koehler, D.O.: A note on some operator theory in certain semi-inner product spaces. Proc. Amer. Math. Soc. 30, 363–366 (1971)

    MathSciNet  MATH  Google Scholar 

  18. Liu, R.: On shrinking and boundedly complete Schauder frames. J. Math. Anal. Appl. 365, 385–398 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lumer, G.: Semi-inner product spaces. Trans. Amer. Math. Soc. 100, 29–43 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  20. Nanda, S.: Numerical range for two non-linear operators in semi-inner product space. J. Nat. Acad. Math. 17, 16–20 (2003)

    MathSciNet  MATH  Google Scholar 

  21. Pap, E., Pavlovic, R.: Adjoint theorem on semi-inner product spaces of type (p). Univ. u Novom Sadu Zb. Prirod.-Mat. Fak. Ser. Mat. 25(1), 39–46 (1995)

    MathSciNet  Google Scholar 

  22. Zhang, H., Zhang, J.: Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products. Appl. Comput. Harmon. Anal. 31, 1–25 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are thankful to the referees for their valuable suggestions which improved the presentation of the paper.

Author information

Authors and Affiliations

  1. Dhirubhai Ambani Institute of Information and Communication Technology, Gandhinagar, 382007, India

    N. K. Sahu

  2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    C. Nahak

  3. Department of Mathematics, University of Central Florida, Orlando, FL, 32816, USA

    Ram N. Mohapatra

Authors
  1. N. K. Sahu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Nahak
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ram N. Mohapatra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. K. Sahu .

Editor information

Editors and Affiliations

  1. Haldia Institute of Technology, Haldia, India

    Debasis Giri

  2. University of Central Florida, Orlando, Florida, USA

    Ram N. Mohapatra

  3. Freie Universität Berlin, Berlin, Germany

    Heinrich Begehr

  4. Fordham University, Bronx, New York, USA

    Mohammad S. Obaidat

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Sahu, N.K., Nahak, C., Mohapatra, R.N. (2017). Bessel Sequences and Frames in Semi-inner Product Spaces. In: Giri, D., Mohapatra, R., Begehr, H., Obaidat, M. (eds) Mathematics and Computing. ICMC 2017. Communications in Computer and Information Science, vol 655. Springer, Singapore. https://doi.org/10.1007/978-981-10-4642-1_14

Download citation

  • DOI: https://doi.org/10.1007/978-981-10-4642-1_14

  • Published:

  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-4641-4

  • Online ISBN: 978-981-10-4642-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics