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Banach Λ-Frames for Operator Spaces

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DOI: 10.4236/apm.2014.48048    3,503 Downloads   3,963 Views   Citations

ABSTRACT

The Banach frame for a Banach space X can reconstruct each vector in X by the pre-frame operator or the reconstruction operator. The Banach Λ-frame for operator spaces was introduced by Kaushik, Vashisht and Khattar [Reconstruction Property and Frames in Banach Spaces, Palestine Journal of Mathematics, 3(1), 2014, 11-26]. In this paper we give necessary and sufficient conditions for the existence of the Banach Λ-frames. A Paley-Wiener type stability theorem for Λ-Banach frames is discussed.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Singh, M. and Chugh, R. (2014) Banach Λ-Frames for Operator Spaces. Advances in Pure Mathematics, 4, 373-380. doi: 10.4236/apm.2014.48048.

References

[1] Duffin, R.J. and Schaeffer, A.C. (1952) A Class of Non-Harmonic Fourier Series. Transactions of the American Mathematical Society, 72, 341-366. http://dx.doi.org/10.1090/S0002-9947-1952-0047179-6
[2] Gabor, D. (1946) Theory of Communications. Journal of the Institute of Electrical Engineers, 93, 429-457.
[3] Daubechies, I., Grossmann, A. and Meyer, Y. (1986) Painless Non-Orthogonal Expansions. Journal of Mathematical Physics, 27, 1271-1283. http://dx.doi.org/10.1063/1.527388
[4] Grochenig, K. (1991) Describing Functions: Atomic Decompositions versus Frames. Monatshefte für Mathematik, 112, 1-41. http://dx.doi.org/10.1007/BF01321715
[5] Feichtinger, H. and Grochenig, K. (1989) Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions I. Journal of Functional Analysis, 86, 307-340.
http://dx.doi.org/10.1016/0022-1236(89)90055-4
[6] Feichtinger, H. and Grochenig, K. (1989) Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions II. Monatshefte für Mathematik, 108, 129-148.
http://dx.doi.org/10.1007/BF01308667
[7] Christensen, O. and Heil, C. (1997) Pertubation of Banach Frames and Atomic Decompositions. Mathematische Nachrichten, 185, 33-47. http://dx.doi.org/10.1002/mana.3211850104
[8] Coifman, R.R. and Weiss, G. (1977) Expansions of Hardy Spaces and Their Use in Analysis. Bulletin of the American Mathematical Society, 83, 569-645. http://dx.doi.org/10.1090/S0002-9904-1977-14325-5
[9] Casazza, P.G., Han, D. and Larson, D.R. (1999) Frames for Banach Spaces. Contemporary Mathematics, 247, 149-182. http://dx.doi.org/10.1090/conm/247/03801
[10] Jain, P.K., Kaushik, S.K. and Vashisht, L.K. (2006) On Banach Frames. Indian Journal of Pure and Applied Mathematics, 37, 265-272.
[11] Jain, P.K., Kaushik, S.K. and Vashisht, L.K. (2007) On Stability of Banach Frames. Bulletin of the Korean Mathematical Society, 44, 73-81. http://dx.doi.org/10.4134/BKMS.2007.44.1.073
[12] Chugh, R., Singh, M. and Vashisht, L.K. (2014) On Λ-Type Duality of Frames in Banach Spaces. International Journal of Analysis and Applications, 4, 148-158.
[13] Vashisht, L.K. (2012) On Retro Banach Frames of Type P. Azerbaijan Journal of Mathematics, 2, 82-89.
[14] Vashisht, L.K. (2012) On Φ-Schauder Frames. TWMS Journal of Applied and Engineering Mathematics (JAEM), 2, 116-120.
[15] Vashisht, L.K. (2012) On Frames in Banach Spaces. Communications in Mathematics and Applications, 3, 313-332.
[16] Vashisht, L.K. and Sharma, S. (2012) On Weighted Banach Frames. Communications in Mathematics and Applications, 3, 283-292.
[17] Vashisht, L.K. and Sharma, S. (2014) Frames of Eigenfunctions Associated with a Boundary Value Problem. International Journal of Analysis, 2014, Article ID: 590324.
http://dx.doi.org/10.1155/2014/590324
[18] Han, D. and Larson, D.R. (2000) Frames, Bases and Group Representations. Memoirs of the American Mathematical Society, 147, 1-91.
[19] Casazza, P.G. and Christensen, O. (2008) The Reconstruction Property in Banach Spaces and a Perturbation Theorem. Canadian Mathematical Bulletin, 51, 348-358. http://dx.doi.org/10.4153/CMB-2008-035-3
[20] Kaushik, S.K., Vashisht, L.K. and Khattar, G. (2014) Reconstruction Property and Frames in Banach Spaces. Palestine Journal of Mathematics, 3, 11-26.
[21] Khattar, G. and Vashisht, L.K. (2014) The Reconstruction Property in Banach Spaces Generated by Matrices. Advances in Pure and Applied Mathematics, 5, 151-160.
[22] Khattar, G. and Vashisht, L.K. (2014) Some Types of Convergence Related to the Reconstruction Property in Banach Spaces. Submitted.
[23] Vashisht, L.K. and Khattar, G. (2013) On -Reconstruction Property. Advances in Pure Mathematics, 3, 324-330.
http://dx.doi.org/10.4236/apm.2013.33046
[24] Casazza, P.G. and Kutynoik, G. (2012) Finite Frames. Birkhauser.
[25] Heil, C. and Walnut, D. (1989) Continuous and Discrete Wavelet Transforms. SIAM Review, 31, 628-666.
http://dx.doi.org/10.1137/1031129
[26] Young, R. (2001) A Introduction to Non-Harmonic Fourier Series. Academic Press, New York.
[27] Casazza, P.G. (2001) Approximation Properties. In: Johnson, W.B. and Lindenstrauss, J., Eds., Handbook on the Geometry of Banach Spaces, Vol. I, 271-316.
[28] Casazza, P.G. and Christensen, O. (1997) Perturbation of Operators and Applications to Frame Theory. Journal of Fourier Analysis and Applications, 3, 543-557. http://dx.doi.org/10.1007/BF02648883

  
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