Advertisement

Bessel Sequences and Frames in Semi-inner Product Spaces

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

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 \(l^{p}\), \(1<p<\infty \).

Keywords

Semi-inner product space Uniformly convex smooth Banach space Bessel sequence Frame 

Notes

Acknowledgements

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

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, Oxford (2000)MATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carando, D., Lassalle, S., Schmidberg, P.: The reconstruction formula for Banach frames and duality. J. Approx. Theory 163, 640–651 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Casazza, P.G., Christensen, O.: Perturbation of operators and applications to frame theory. J. Four. Anal. Appl. 3, 543–557 (1997)MathSciNetCrossRefMATHGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Casazza, P.G., Christensen, O., Stoeva, D.T.: Frame expansions in separable Banach spaces. J. Math. Anal. Appl. 307(2), 710–723 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Christensen, O.: Frames and Bases: An Introductory Course. Birkhauser, Boston (2008)CrossRefMATHGoogle Scholar
  9. 9.
    Christensen, O.: Frames and pseudo-inverse operators. J. Math. Anal. Appl. 195, 401–414 (1995)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Christensen, O.: Frames perturbations. Proc. Amer. Math. Soc. 123, 1217–1220 (1995)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Christensen, O.: A Paley-Wiener theorem for frames. Proc. Amer. Math. Soc. 123, 2199–2202 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nachr. 185, 33–47 (1997)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Favier, S.J., Zalik, R.A.: On the stability of frames and Riesz bases. Appl. Compu. Harm. Anal. 2, 160–173 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Fornasier, M.: Banach frames for \(\alpha \)-modulation spaces. Appl. Comput. Harmon. Anal. 22, 157–175 (2007)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giles, J.R.: Classes of semi-inner product spaces. Trans. Amer. Math. Soc. 129, 436–446 (1967)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Grochenig, K.: Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Koehler, D.O.: A note on some operator theory in certain semi-inner product spaces. Proc. Amer. Math. Soc. 30, 363–366 (1971)MathSciNetMATHGoogle Scholar
  18. 18.
    Liu, R.: On shrinking and boundedly complete Schauder frames. J. Math. Anal. Appl. 365, 385–398 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lumer, G.: Semi-inner product spaces. Trans. Amer. Math. Soc. 100, 29–43 (1961)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Nanda, S.: Numerical range for two non-linear operators in semi-inner product space. J. Nat. Acad. Math. 17, 16–20 (2003)MathSciNetMATHGoogle Scholar
  21. 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)MathSciNetGoogle Scholar
  22. 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)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Dhirubhai Ambani Institute of Information and Communication TechnologyGandhinagarIndia
  2. 2.Department of MathematicsIndian Institute of Technology KharagpurKharagpurIndia
  3. 3.Department of MathematicsUniversity of Central FloridaOrlandoUSA

Personalised recommendations