Document Type : Research Paper
Author
Department of Mathematics, Kalyan P.G. College, Bhilai Nagar (C.G.) 490006 India.
Abstract
A new class of retro Banach frames called retro bi-Banach frame has been introduced and studied with illustrative examples. Relationships of a retro bi-Banach frame with various existing classes of Banach frame are presented. In the sequel, we deal with characterizations of the near-exact retro Banach frame and discuss the invariance of near-exact retro Banach frames under block perturbation. Finally, applications regarding the rank of a matrix and eigenvalue problems have been demonstrated.
Keywords
[1] M.R. Abdollahpour, M.H. Faroughi and and A. Rahimi, $PG$-frames in Banach spaces, Methods Function. Anal. Topology, 13 (3), (2007), pp. 201-210.
[2] A. Aldroubi, Q. Sun and W. Tang, $p$-frames and invariant subspaces of $L^p$, J. Fourier Anal. Appl., 7 (1), (2001), pp. 1-21.
[3] P.G. Casazza, D. Han and D. R. Larson, Frames for Banach spaces, Contemp. Math. 247, (1999), pp. 149-182.
[4] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., 345 (Amer. Math. Soc., Providence, RI, 2004), pp. 87-113.
[5] P.G. Casazza, O. Christensen and D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl., 307 (2005), pp. 710-723.
[6] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston, (2003).
[7] O. Christensen, Frames and bases, an introductory course, Birkhauser, Boston, (2008).
[8] R.J. Duffin and A.C. Scheaffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.
[9] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986), pp. 1271-1283.
[10] H.G. Feichtinger and K. Grochenig, A unified approach to atomic decompositions, via integrable group representations, Lect. Notes Math., 1302 (Berlin-Heidelberg-New York, Springer, 1988), pp. 52-73.
[11] D. Gabor, Theory of communications, J. Inst. Elec. Engg., 93 (1946), pp. 429-457.
[12] M.P. Goswami and H.K. Pathak, Some results on $\Lambda$-Banach frames for operator spaces, Jordan J. Math. Stast., 11 (2) (2018), pp. 169-194.
[13] K. Grochenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math., 112 (1991), pp. 1-41.
[14] M.I. Ismailov and A. Jabrailova, On $\tilde{X}$-frames and conjugate systems in Banach spaces, Sahand Commun. Math. Anal., 1 (2) (2014), pp. 19-29.
[15] S. Jahan, V. Kumar and S.K. Kaushik, On the existence of non-ninear frames, Arch. Math., 53 (2017), pp. 101-109.
[16] S. Jahan and V. Kumar, Some Results on the existence of frames in Banach spaces, Poincare J. Anal. Appl., (1) (2018) , pp. 25-33.
[17] P.K. Jain, S.K. Kaushik and L.K. Vashisht, Banach frames for conjugate Banach spaces, Zeit. Anal. Anwendungen, 23 4 (2004), pp. 713-720.
[18] P.K. Jain, S.K. Kaushik, L.K. Vashisht, On Banach frames, Indian J. Pure and Appl. Math., 37 5 (2006), pp. 265-272.
[19] P.K. Jain, S.K. Kaushik and N. Gupta, On near-exact Banach frames in Banach spaces, Bull. Austral.Math. Soc., 78 (2008), pp. 335-342.
[20] P.K. Jain, S.K. Kaushik and N. Gupta, On frame systems in Banach spaces, Int. J. Wavelets, Multiresolut. Inf. Process., 7 1 (2009), pp. 1-7.
[21] N. Narayan jha and S. Shrama, Block sequences of retro banach frames, Poincare J. Anal. Appl., 7 (2) (2020), pp. 267-274.
[22] S.K. Kaushik, Some results concerning frames in Banach spaces, Tamkang J. Math., 38 (3) (2007), pp. 267-276.
[23] M. Rashidi Kouchi and A. Nazari, Some relationships between G-frames and frames, Sahand Commun. Math. Anal., 2(1) (2015), pp. 1-7.
[24] R. Kumar and S.K. Sharma, A note on retro Banach frames, Int. J. Pure Appl. math., 48 1 (2008), pp. 111-115.
[25] K.T. Poumai and S. Jahan, On K-atomic decompositions in Banach Spaces, Electron. J. Math. Anal. Appl., 6 (1) (2018), pp. 183-197.
[26] A. Rahimi, Invariance of Frechet frames under perturbation, Sahand Commun. Math. Anal., 1 (1) (2014), pp. 41-51.
[27] S. Sharma, On bi-Banach frames in Banach spaces, Int. J. Wavelets, Multiresolut. Inf. Process., 12 2 (2014), 10 pages.
[28] I. Singer, Bases in Banach spaces II, Springer, New York, 1981.
[29] W.C. Sun, $G$-frames and $G$-Riesz Bases, J. Math. Anal. Appl., 322 (2006), pp. 437-452.
[30] R. Young, An introduction to non-harmonic Fourier series, Pure and Appl. Math., Academic press, New York, NY, USA, 93, 1980.
)