APM  Vol.3 No.3 , May 2013
On З-Reconstruction Property
Reconstruction property in Banach spaces introduced and studied by Casazza and Christensen in [1]. In this paper we introduce reconstruction property in Banach spaces which satisfy -property. A characterization of reconstruction property in Banach spaces which satisfy -property in terms of frames in Banach spaces is obtained. Banach frames associated with reconstruction property are discussed.

Cite this paper
L. Vashisht and G. Khattar, "On З-Reconstruction Property," Advances in Pure Mathematics, Vol. 3 No. 3, 2013, pp. 324-330. doi: 10.4236/apm.2013.33046.
[1]   P. G. Casazza and O. Christensen, “The Reconstruction Property in Banach Spaces and a Perturbation Theorem,” Canadian Mathematical Bulletin, Vol. 51, No. 3, 2008, pp. 348-358. doi:10.4153/CMB-2008-035-3

[2]   R. J. Duffin and A. C. Schaeffer, “A Class of Non-Harmonic Fourier Series,” Transactions of the American Mathematical Society, Vol. 72, No. 2, 1952, pp. 341-366. doi:10.1090/S0002-9947-1952-0047179-6

[3]   D. Gabor, “Theory of Communications. Part 1: The Analysis of Information,” Journal of the Institution of Electrical Engineers, Vol. 93, No. 26, 1946, pp. 429-457.

[4]   I. Daubechies, A. Grossmann and Y. Meyer, “Painless Non-Orthogonal Expansions,” Journal of Mathematical Physics, Vol. 27, No. 5, 1986, pp. 1271-1283. doi:10.1063/1.527388

[5]   K. Grochenig, “Describing Functions: Atomic Decompositions versus Frames,” Monatshefte für Mathematik, Vol. 112, No. 1, 1991, pp. 1-42. doi:10.1007/BF01321715

[6]   H. G. Feichtinger and K. Grochenig, “Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions, I,” Journal of Functional Analysis, Vol. 86, No. 2, 1989, pp. 305-340. doi:10.1016/0022-1236(89)90055-4

[7]   H. G. Feichtinger and K. Grochenig, “Banach Spaces Related to Integrable Group Representations and Their Atomic Decompositions, II,” Monatshefte für Mathematik, Vol. 108, No. 2-3, 1989, pp. 129-148. doi:10.1007/BF01308667

[8]   O. Christensen and C. Heil, “Perturbation of Banach Frames and Atomic Decomposition,” Mathematische Nachrichten, Vol. 185, No. 1, 1997, pp. 33-47. doi:10.1002/mana.3211850104

[9]   R. R. Coifman and G. Weiss, “Extensions of Hardy Spaces and Their Use in Analysis,” Bulletin of the American Mathematical Society, Vol. 83, No. 4, 1977, pp. 569-645. doi:10.1090/S0002-9904-1977-14325-5

[10]   R. Young, “A Introduction to Non-Harmonic Fourier Series,” Academic Press, New York, 1980.

[11]   I. Daubechies, “Ten Lectures on Wavelets,” Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1992. doi:10.1137/1.9781611970104

[12]   C. Heil and D. Walnut, “Continuous and Discrete Wavelet Transforms,” SIAM Review, Vol. 31, No. 4, 1989, pp. 628-666. doi:10.1137/1031129

[13]   O. Christensen, “Frames and Bases (An Introductory Course),” Birkhauser, Boston, 2008.

[14]   P. K. Jain, S. K. Kaushik and L. K. Vashisht, “Banach Frames for Conjugate Banach Spaces,” Zeitschrift für Analysis und ihre Anwendungen, Vol. 23, No. 4, 2004, pp. 713-720. doi:10.4171/ZAA/1217

[15]   P. G. Casazza, D. Han and D. R. Larson, “Frames for Banach Spaces,” Contemporary Mathematics, Vol. 247, 1999, pp. 149-182. doi:10.1090/conm/247/03801

[16]   L. K. Vashisht, “On Φ-Schauder Frames,” TWMS Journal of Applied and Engineering Mathematics (JAEM), Vol. 2, No. 1, 2012, pp. 116-120.

[17]   M. A. Neumark, “Linear Differential Operator (Translation from the Russian),” Akademie-Verlag, Berlin, 1960.

[18]   F. Riesz and B.Sz.-Nagy, “Functional Analysis,” F. Ungar Co., New York, 1955.