Volume 69 | Issue 1 | Year 2023 | Article Id. IJMTT-V69I1P510 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I1P510
Received | Revised | Accepted | Published |
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03 Dec 2022 | 05 Jan 2023 | 17 Jan 2023 | 31 Jan 2023 |
In this research, the retro Banach frames are presented and different conditions are developed for the applications in the complex conjugate spaces. The retro Banach spaces are separable Banach spaces and used for the signal processing in the conjugate complex Banach spaces. Conventionally, the retro Banach spaces are applied to the Hilbert transform to be utilized in the conjugate Banach space. This research presents the further conditions for retro Banach frames that can be applied to the Wavelet and Gober’s transform such that this transform can be used in the conjugate complex Banach spaces. The current research discusses facts related to the form of Retro Banach Frames, Conjugate Banach Spaces, Linear Isomerism, Linear monomorphism, and Schauder basiss. It also helps in executing, extending, and modifying the Schauder frames to give a perfect characterization over the hesitation of Schauder frames.
[1] Peter G. Casazza, “Finite Dimensional Decompositions in Banach Spaces,” Contemporary Mathematics, vol. 52, pp. 1–31, 1986.
[2] Peter G. Casazza, Deguang Han, and David R. Larson, “Frames for Banach Spaces,” Contemporary Mathematics, pp. 1-34, 1999.
[3] Peter G. Casazza, and Gitta Kutyniok, “Frames of Subspaces,” Contomporary Mathematics, pp. 1-27, 2004.
[4] Ole Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston Inc, Boston MA.
[5] Ole Christensen, and Christopher Heil, “Perturbations of Banach Frames and Atomic Decompositions,” Mathematical News, vol. 185, pp. 33-47, 1997.
[6] Ronald R. Coifman, and Guido Weiss, “Extensions of Hardy Spaces and Their use in Analysis,” Bulletin of the American Mathematical Society, vol. 83, pp. 569-645, 1977.
[7] Richard Duffin, and A.C. Schaeffer, “Some Properties of Functions of Exponential Type,” Bulletin of the American Mathematical Society, vol. 44, no. 4, pp. 236-240, 1938.
[8] R.J. Duffin, and A.C. Schaeffer, “ A Class of Nonharmonic Fourier Series,” Transaction of the American Mathematical Society, vol. 72, no. 2, pp. 341-366, 1952. Crossref, https://doi.org/10.2307/1990760
[9] Laura Gavruta, “Frames for Operators,” Applied and Computational Harmonic Analysis, vol. 32, no. 1, pp. 139-144, 2012. Crossref, https://doi.org/10.1016/j.acha.2011.07.006
[10] Laura Gavruta, “New Results on Frame for Operators,” Annals of Oradea University-Mathematics Fascicola, vol. 2, pp. 55-61, 2012.
[11] P. Gavruta, “On the Duality of Fusion Frames,” Journal of Mathematical Analysis and Applications, vol. 333, no. 2, pp. 871- 879, 2007. Crossref, https://doi.org/10.1016/j.jmaa.2006.11.052
[12] Deguang Han, and David R. Larson, “Frames, Bases and Group Representations,” Memoirs of the American Mathematical Society, vol. 147, 2000.
[13] Christopher Heil, “A Basis Theory Primer,” Lecture Notes, Georgia.
[14] P.K. Jain, Lalit K. Vashisht, and S.K. Kaushik, “Banach Frames for Conjugate Banach Spaces,” Journal for Analysis and its Applications, vol. 23, no. 4, pp. 713–720, 2004. Crossref, https://doi.org/10.4171/zaa/1217
[15] P.K. Jain, Shiv Kumar Kaushik, and Lalit Kumar Vashisht, “On Banach Frames,” Indian Journal of Pure and Applied Mathematics, vol. 37, no. 5, 2006.
[16] P.K. Jain, S.K. Kaushik, and Varinder Kumar, “Frames of Subspaces for Banach Spaces,” International Journal of Wavelets Multiresolution and Information Processing, vol. 8, no. 2, pp. 243- 252, 2010. Crossref, https://doi.org/10.1142/S0219691310003481
[17] P.K. Jain, S.K. Kaushik, and L.K. Vashisht, “On Perturbation of Banach frames,” International Journal of Wavelets Multiresolution and Information Processing, vol. 4, no. 3, pp. 559- 565, 2006. Crossref, https://doi.org/10.1142/S0219691306001440
[18] P.K. Jain, S.K. Kaushik, and Nisha Gupta, “On Banach Frame Systems in Banach spaces,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 7, no. 1, pp. 1-7, 2009. Crossref, https://doi.org/10.1142/S021969130900274X
[19] S.K. Kaushik, “Some Results Concerning Frames in Banach Spaces,” Tamkang Journal of Mathematics, vol. 38, no. 3, pp. 267–276, 2007. Crossref, https://doi.org/10.5556/j.tkjm.38.2007.80
[20] S.K. Kaushik, “A Generalization of Frames in Banach Spaces,” Journal of Contemporary Mathematical Analysis, vol. 44, pp. 212-218, 2009. Crossref, https://doi.org/10.3103/S1068362309040025
[21] S.K. Kaushik, and S.K. Sharma, “On Atomic Decompositions Satisfying Certain Properties,” Journal of Advanced Research in Pure Mathematics, vol. 3, no. 2, pp. 40–48, 2011.
[22] S.K. Kaushik, and S.K. Sharma, “On Approximative Atomic Decompositions in Banach spaces,” Communications in Mathematics and Applications, vol. 3, no. 3, pp. 293-301, 2012. Crossref, https://doi.org/10.26713/cma.v3i3.213
[23] Rui Liu, “On Shrinking and Boundedly Complete Schauder Frames for Banach Space,” Journal of Mathematical Analysis and Applications, vol. 365, no. 1, pp. 385-398, 2010. Crossref, https://doi.org/10.1016/j.jmaa.2009.11.001
[24] Rui Liu, and Bentuo Zheng, “A Characterization of Schauder Frames which are Near- Schauder Bases,” Journal of Fourier Analysis and Applications, vol. 16, no. 5, pp. 1-12, 2010. Crossref, http://dx.doi.org/10.1007/s00041-010-9126-5
[25] Shidong Li, “On General Frame Decompositions,” Numerical Functional Analysis and Optimization, vol. 16, no. 9-10, pp. 1181–1191, 1995. Crossref, https://doi.org/10.1080/01630569508816668
[26] G. Verma, L.K. Vashisht, and M. Singh, “Excess of Retro Banach Frames,” Journal of Contemporary Mathematical Analysis, vol. 54, pp. 143–146, 2019. Crossref, https://doi.org/10.3103/S1068362319030026
[27] Chander Shekhar, and Ghanshyam Singh Rathore, “Retro K-banach Frames in Banach Spaces,” Poincare Journal of Analysis and Applications, vol. 2, no. 1, pp. 65-75, 2018.
[28] Lalit Kumar Vashisht, “On retro Banach Frames of Type P,” Azerbaijan Journal of Mathematics, vol. 2, no. 1, 2012.
Shipra, "The Retro Banach Frames and their Applications in Conjugate Banach Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 1, pp. 67-70, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I1P510