Skip to main content
SpringerLink
Log in

Frames in Semi-inner Product Spaces

  • Conference paper
  • First Online:
Mathematical Analysis and its Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 143))

Abstract

The objective of this paper is to study the theory of frames in semi-inner product spaces. Several researchers have studied frames in Banach spaces by using the bounded linear functionals. Application of semi-inner product is a new approach to investigate the theory of frames. The notion of semi-frame is introduced in this new aspect.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Antoine, J.P., Balazs, P.: Frames and semi-frames. J. Phys. A: Math. Theor. 44(20), 205201 (2011)

    Article  MathSciNet  Google Scholar 

  2. Casazza, P.G., Christensen, O.: The reconstruction property in Banach spaces and perturbation theorem. Canad. Math. Bull. 51, 348–358 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhauser, Boston (2002)

    Google Scholar 

  4. Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nachr. 158, 33–47 (1997)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Gröchenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Four. Anal. Appl. 10, 105–132 (2004)

    Article  MATH  Google Scholar 

  8. Han, D., Jing, W., Mohapatra, R.N.: Structured Parseval frames in Hilbert \(C^{*}\) modules. Contemp. Math. (American Math. Soc.) 414, 275–288 (2006)

    Google Scholar 

  9. Han, D., Jing, W., Larson, D., Mohapatra, R.N.: Riesz bases and their dual modular frames in Hilbert \(C^{*}\) modules. J. Math. Anal. Appl. 343, 246–256 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Han, D., Jing, W., Mohapatra, R.N.: Perturbation of frames and Riesz bases in Hilbert \(C^{*}\) modules. Linear Algebra Appl. 431, 746–759 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Han, D., Jing, W., Larson, D., Li, P., Mohapatra, R.N.: Dilation of dual frame pairs in Hilbert \(C^{*}\) modules. Resultate der Mathematik 63, 241–250 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Han, D., Sun, W.: Reconstruction of signals from frame coefficients with erasures at unknown locations. IEEE Trans. Inform. Theor. 60(7), 4013–4025 (2014)

    Article  MathSciNet  Google Scholar 

  13. Kaushik, S.K.: A generalization of frames in Banach spaces. J. Contemp. Math. Anal. 44, 212–218 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Pehlivan, S., Han, D., Mohapatra, R.N.: Linearly connected sequences and spectrally optimal dual frames for erasures. J. Funct. Anal. 265(11), 2855–2876 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Sahu, N.K., Nahak, C.: Semi-inner product: application to frame theory and numerical range of operators. Springer Proceedings in Mathematics & Statistics, pp. 77–90. Springer, New York (2014)

    Google Scholar 

  17. Stoeva, D.T.: On \(p\)-frames and reconstruction series in separable Banach spaces. Integ. Transforms Spec. Functions 17, 127–133 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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

Acknowledgments

We thank the referees for their comments which have been used in the final presentation.

Author information

Authors and Affiliations

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

    N. K. Sahu

  2. 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. 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. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    P. N. Agrawal

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

    R. N. Mohapatra

  3. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, India

    Uaday Singh

  4. Dept. of Mathematics and Statistics, University of Victoria, Victoria, British Columbia, Canada

    H. M. Srivastava

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer India

About this paper

Cite this paper

Sahu, N.K., Mohapatra, R.N. (2015). Frames in Semi-inner Product Spaces. In: Agrawal, P., Mohapatra, R., Singh, U., Srivastava, H. (eds) Mathematical Analysis and its Applications. Springer Proceedings in Mathematics & Statistics, vol 143. Springer, New Delhi. https://doi.org/10.1007/978-81-322-2485-3_11

Download citation

Publish with us

Policies and ethics

Search

Navigation