Abstract: We introduce the notion of a retro Banach frame relative to a bounded \(b\)-linear functional in \(n\)-Banach space and see that the sum of two retro Banach frames in \(n\)-Banach space with different reconstructions operators is also a retro Banach frame in \(n\)-Banach space. Also, we define retro Banach Bessel sequence with respect to a bounded \(b\)-linear functional in \(n\)-Banach space. A necessary and sufficient condition for the stability of retro Banach frame with respect to bounded \(b\)-linear functional in \(n\)-Banach space is being obtained. Further, we prove that retro Banach frame with respect to bounded \(b\)-linear functional in \(n\)-Banach space is stable under perturbation of frame elements by positively confined sequence of scalars. In \(n\)-Banach space, some perturbation results of retro Banach frame with the help of bounded \(b\)-linear functional in \(n\)-Banach space have been studied. Finally, we give a sufficient condition for finite sum of retro Banach frames to be a retro Banach frame in \(n\)-Banach space. At the end, we discuss retro Banach frame with respect to a bounded \(b\)-linear functional in Cartesian product of two \(n\)-Banach spaces.
For citation: Ghosh, P. and Samanta, T. K. On Stability of Retro Banach Frame with Respect to \(b\)-Linear Functional in \(n\)-Banach Space, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 48-63. DOI 10.46698/o3961-3328-9819-i
1. Gabor, D. Theory of Communications,
Journal of Institution of Electrical Engineers,
1946, vol. 93, pp. 429-457.
2. Duffin, R. J. and Schaeffer, A. C. A Class of Nonharmonic Fourier Series,
Transactions of the American Mathematical Society,
1952, vol. 72, no. 2, pp. 341-366.
DOI: 10.1090/s0002-9947-1952-0047179-6.
3. Daubechies, I., Grossmann, A. and Mayer, Y.
Painless Nonorthogonal Expansions, Journal of Mathematical Physics,
1986, vol. 27, no. 5, pp. 1271-1283. DOI: 10.1063/1.527388.
4. Feichtinger, H. G. and Grochenig, K.
Banach Spaces Related to Integrable Group Representation
and their Atomic Decompositions, I,
Journal of Functional Analysis, 1989, vol. 86, no. 2, pp. 307-340.
DOI: 10.1016/0022-1236(89)90055-4.
5. Feichtinger, H. G. and Grochenig, K.
Banach Spaces Related to Integrable Group Representation and their Atomic Decompositions, II,
Monatshefte fur Mathematik, 1989, vol. 108, pp. 129-148.
DOI: 10.1007/BF01308667.
6. Grochenig, K. Describing Functions: Atomic Decomposition Versus Frames,
Monatshefte fur Mathematik, 1991, vol. 112, no. 1, pp. 1-42.
DOI: 10.1007/BF01321715.
7. Cazassa, P. G., Han, D. and Larson, D. R. Frames for Banach Spaces,
The Functional and Harmonic Analysis of Wavelets and Frames,
Contemporary Mathematics, vol. 247, American Mathematical Society,
Providence, R.I., 1999, pp. 149-182. DOI: 10.1090/conm/247/03801.
8. Jain, P. K., Kaushik, S. K. and Vashisht, L. K.
Banach Frames for Conjugate Banach Spaces,
Journal for Analysis and its Applications, 2004, vol. 23, no. 4,
pp. 713-720.DOI: 10.4171/ZAA/1217.
9. Vashisht, L. K. On Retro Banach Frames of Type P,
Azerbaijan Journal of Mathematics, 2012, vol. 2, pp. 82-89.
10. Christensen, O. and Heil, C. Perturbation of Banach Frames and Atomic Decompositions,
Mathematische Nachrichten, 1997, vol. 185, no. 1, pp. 33-47.
DOI: 10.1002/mana.3211850104.
11. Jain, P. K., Kaushik, S. K. and Vashisht, L. K.
On Stability of Banach Frames, Bulletin of the Korean Mathematical Society,
2007, vol. 44, no. 1, pp. 73-81. DOI: 10.4134/bkms.2007.44.1.073.
12. Gahler, S. Lineare 2-Normierte Raume,
Mathematische Nachrichten, 1964, vol. 28, no. 1-2, pp. 1-43.
DOI: 10.1002/mana.19640280102.
13. Gunawan, H. and Mashadi, M. On \(n\)-Normed Spaces,
International Journal of Mathematics and Mathematical Sciences,
2001, vol. 27, no. 10, pp. 631-639. DOI: 10.1155/s0161171201010675.
14. Ghosh, P. and Samanta, T. K. Construction of Frame Relative
to \(n\)-Hilbert Space, Journal of Linear and Topological Algebra,
2021, vol. 10, no. 02, pp. 117-130.
15. Ghosh, P. and Samanta, T. K. Introduction of Frame in Tensor Product
of \(n\)-Hilbert Spaces, Sahand Communication in Mathematical Analysis, 2021,
vol. 18, no. 4, pp. 1-18. DOI: 10.22130/scma.2021.524252.909.
16. Ghosh, P. and Samanta, T. K. Atomic Systems in \(n\)-Hilbert
Spaces and their Tensor Products,
Journal Linear Topological Algebra, 2021,
vol. 10, no. 04, pp. 241-256.
17. Ghosh, P. and Samanta, T. K. Representation of Uniform Boundedness
Principle and Hahn-Banach Theorem in Linear \(n\)-Normed Space,
The Journal of Analysis, 2022, vol. 30, no. 2, pp. 597-619.
DOI: 10.1007/s41478-021-00358-x.
18. Singer, I. Bases in Banach Spaces. II,
New York-Heidelberg, Springer-Verlag, 1981.
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