Generalized pseudo $B$-Gabor frames on finite abelian groups
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Abstract
We seek for an invertible map $B$ from $L^2(\Gamma)$ to $L^2(G)$, where $G$ is a finite abelian group and $\Gamma$ is the direct product of finite cyclic groups which is isomorphic to $G$, so that any Gabor frame in $L^2(G)$, is a generalized pseudo $B$-Gabor frame.
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References
[1] P. G. Cazassa and G. Kutyniok, Finite Frames Theory and Applications, Applied and Numerical Harmonic Analysis Series, Birkh ̈auser, Boston, 2013. https://doi.org/10.1007/978-0-8176-8373-3 Google Scholar
[2] O. Christensen, An Introduction to Frames and Riesz Bases, Second Edition, Birkh ̈auser, Boston, 2016. Google Scholar
[3] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27 (1986), 1271-1283. https://doi.org/10.1063/1.527388 Google Scholar
[4] R. J. Duffin and A. C. Schaeffler, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. https://doi.org/10.1090/S0002-9947-1952-0047179-6 Google Scholar
[5] T. C. Easwaran Nambudiri, K. Parthasarathy, Generalized Weyl-Heisenberg frame operators, Bull. Sci. Math., 136 (2012), 44–53. https://doi.org/10.1016/j.bulsci.2011.09.001 Google Scholar
[6] T. C. Easwaran Nambudiri, K. Parthasarathy, Characterization of Weyl-Heisenberg frame op- erators, Bull. Sci. Math., 137 (2013), 322–324. https://doi.org/10.1016/j.bulsci.2012.09.001 Google Scholar
[7] H. G. Feichtinger, W. Kozek, Quantization of TF–lattice invariant operators on elementary LCA groups, In H.G. Feichtinger, T. Strohmer (eds.) Gabor Analysis and Algorithms: Theory and Applications, Birkh ̈auser, Boston, MA (1998), 233–266. Google Scholar
[8] H. G. Feichtinger, W. Kozek, Luef F, Gabor analysis over finite abelian groups, Appl. Comput. Harmon. Anal. 26 (2) (2009), 230–248. https://doi.org/10.1016/j.acha.2008.04.006 Google Scholar
[9] D. Gabor, Theory of communication, J. IEE, 93 (1946), 429–457. Google Scholar
[10] K. Gro ̈chenig, Foundations of Time Frequency Analysis, Birkh ̈auser, Boston, 2001. Google Scholar
[11] M. Janssen, Gabor representation of generalized functions, J. Math. Anal. Appl., 83 (1981), Google Scholar
[12] –394. https://doi.org/10.1016/0022-247X(81)90130-X Google Scholar
[13] K. D. Lamb, Gerry, C. Christopher, and Grobe, Rainer, Unitary and nonunitary approaches in quantum field theory, (2007). Faculty publications – Physics. 40. https://ir.library.illinoisstate.edu/fpphys/40 Google Scholar
[14] G. Pfander, Gabor frames in finite dimensions, Birkh ̈auser, Boston, 2010. Google Scholar
[15] Terras. A, Fourier analysis on finite groups and applications, London Mathematical Society Google Scholar
[16] Student Texts, vol. 43. Cambridge University Press, Cambridge (1999). Google Scholar
[17] J. Thomas, N. M. M. Namboothiri and T. C. E. Nambudiri, A class of structured frames in finite dimensional hilbert spaces, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 28 (4), (2022), 321–334. https://doi.org/10.7468/jksmeb.2022.29.4.321 Google Scholar