Construction of generalized atomic decompositions in Banach spaces

  • Authors

    • Raj Kumar Department of Mathematics Kirori mal college University of Delhi Delhi-110007 India.
    • Mahesh Joshi Professor DEPARTMENT OF MATHEMATICS, D.S.B,CAMPUS ,Kumaun University. NAINITAL-263001, INDIA
    • Ram Singh DEPARTMENT OF MATHEMATICS, D.S.B,CAMPUS ,Kumaun University. NAINITAL-263001, INDIA
    • Ashok Sah Department of Mathematics Kirori mal college University of Delhi Delhi-110007 India.
  • G-atomic decompositions for Banach spaces with respect to a model space of sequences have been introduced and studied as a generalization of atomic decompositions. Examples and counter example have been provided to show its existence. It has been proved that an associated Banach space for G-atomic decomposition always has a complemented subspace. The notion of a representation system is introduced and exhibits its relation with G-atomic decomposition. Also It has been observed that G-atomic decompositions are exactly compressions of Schauder decompositions for a larger Banach space. We give a characterization for finite G-atomic decomposition in terms of finite-dimensional expansion of identity.

    Keywords: complemented coefficient spaces, finite-dimensional expansion of identity, G-atomic decomposition, representation system.

  • References

    1. D. Carando and S. Lassalle, Duality, refexivity and atomic decompositions in Banach spaces, Studia Math., 191(1) (2009), 67-80.
    2. P.G. Casazza, O. Christensen and D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl., 307 (2005), 710-723.
    3. P.G. Casazza, D. Han and D.R. Larson, Frames for Banach spaces, Contem. Math., 247 (1999), 149-181.
    4. P.G. Casazza and G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Operator Theory (College Park, MD, 2003), Contemp. Math., 345, Amer. Math. Soc., Providence, RI, 2004, 87-113.
    5. P.G. Casazza, G. Kutyniok and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25 (2008), 114-132.
    6. O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math., 26(4) (1996), 1289-1312.
    7. O. Christensen and Y. C. Eldar, Oblique dual frames with shift-invariant spaces, Appl. Compt. Harmon. Anal., 17(1) (2004), 48-68.
    8. O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr., 185 (1997), 33-47.
    9. R.R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645.
    10. H.G. Feichtinger, Atomic characterizations of modulations spaces through Gabor-type representation, Rocky Mountain J. Math., 19(1) (1989), 113-125.
    11. H.G. Feichtinger and K.H. GrÄochenig, A unified approach to atomic decompositions via integrable group representations, Function spaces and Applications, Lecture Notes in Mathematics, 1302 (1988), 52-73.
    12. M. Frazier and B. Jawerth, Decompositions of Besov spaces, Indiana Univ. Math. J., 34 (1985), 777-799.
    13. K.H. GrÄochenig, Describing functions: Atomic decompositions versus frames, Monatsh. fur Mathematik, 112(3) (1991), 1-41.
    14. S.K. Kaushik and Varinder Kumar, A note on fusion Banach frames, Archivum mathematicum(BRNO), Tomus 46 (2010), 203-209.
    15. S. Li and H. Ogawa, Pseudo frames for subspaces with applications, J. Fourier Anal. Appl., 10(4) (2004), 409-431.
    16. A. Pelczynski, Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis, Studia Math., 40(1971), 239-242.
    17. A. Pelczynski and P. Wojtaszczyk, Banach spaces with finite dimensional expansion of identity and universal bases of finite dimensional subspaces, Studia Math., 40 (1971), 91-108.
    18. B.L. Sanders, On the existence of Schauder decomposition in Banach spaces, Proc. Amer. Math. Soc., 16(1965), 987-990.
    19. I. Singer, Bases in Banach spaces. II, Springer (Berlin, 1981).
    20. W. Sun, G-frames and g-Riesz bases, J. Math. Anal. Appl., 322(1) (2006), 437-452.
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  • How to Cite

    Kumar, R., Joshi, M., Singh, R., & Sah, A. (2014). Construction of generalized atomic decompositions in Banach spaces. International Journal of Advanced Mathematical Sciences, 2(3), 116-124.