On Λ-Type Duality of Frames in Banach Spaces

Article Sidebar

PDF
Cite as:
Renu Chugh, Mukesh Singh, L. K. Vashisht, On Λ-Type Duality of Frames in Banach Spaces, Int. J. Anal. Appl., 4 (2) (2014), 148-158.

Main Article Content

Renu Chugh, Mukesh Singh, L. K. Vashisht

Abstract

Frames are redundant system which are useful in the reconstruction of certain classes of spaces. The dual of a frame (Hilbert) always exists and can be obtained in a natural way. Due to the presence of three Banach spaces in the definition of retro Banach frames (or Banach frames) duality of frames in Banach spaces is not similar to frames for Hilbert spaces. In this paper we introduce the notion of Λ-type duality of retro Banach frames. This can be generalized to Banach frames in Banach spaces. Necessary and sufficient conditions for the existence of the dual of retro Banach frames are obtained. A special class of retro Banach frames which always admit a dual frame is discussed.

Article Details

References

  1. P.G.Casazza and G. Kutynoik, Finite Frames, Birkh ¨auser, 2012.
  2. P.G. Casazza, The art of frame theory, Tawanese J. Math., 4(2) (2000), 129-201
  3. P.G. Casazza, D. Han and D.R. Larson, Frames for Banach spaces, Contemp. Math., 247 (1999), 149-182.
  4. P.G.Casazza, G.Kutyniok and M.C.Lammers, Duality principles in frame thoery, J. Fourier Anal. Appl., 10 (2004), 383-408.
  5. O. Christensen, Frames and bases (An introductory course), Birkh ¨auser, Boston (2008).
  6. I.Daubechies, A. Grossmann and Y. Meyer, Painless non-orthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283.
  7. R.J. Duffin and A.C. Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341-366.
  8. I.Daubechies, H. Landau and Z.Landau, Gabor time-frequency lattices and the Wexler-Raz identity, J. Fourier Anal. Appl., 1(4) (1995), 437-478.
  9. H.G. Feichtinger and K. Gr ¨ochenig, A unified approach to atomic decompositons via inegrable group representations, Lecture Notes in Mathematics, 1302 (Springer, Berlin, 1988), 52-73.
  10. K. Gr ¨ochenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math., 112, (1991), 1-41.
  11. D. Han and D.R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc., 147 (697) (2000), 1-91.
  12. C.Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31 (4) (1989), 628-666.
  13. C. Heil, A basis theory primer, Birkh ¨auser (expanded edition)(1998).
  14. H.Heuser, Functional Analysis, John Wiley and Sons, New York (1982).
  15. P.K.Jain, S.K.Kaushik and L.K. Vashisht, Banach frames for conjugate Banach spaces, Z. Anal. Anwendungen, 23 (4) (2004), 713-720.
  16. J. Kovacˇcevi ´c and A. Chebira, Life Beyond Bases: The advent of frames (Part I), IEEE Signal Processing Magazine, 86, July, 2007.
  17. J. Kovacˇcevi ´c and A. Chebira, Life Beyond Bases: The advent of frames (Part II), IEEE Signal Processing Magazine, 115, September, 2007.
  18. L.K. Vashisht, On retro Banach frames of type P, Azerb. J. Math., 2 (1) (2012), 82-89.
  19. L.K. Vashisht, On Φ-Schauder frames, TWMS J. App. and Eng. Math.(JAEM), 2 (1) (2012), 116-120.
  20. R. Young, On complete biorthogonal systems, Proc. Amer. Math. Soc., 83 (3) (1981), 537- 540.
  21. R. Young, A introduction to non-harmonic Fourier series, Academic Press, New York (revised first edition 2001).