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Gabor frames on local fields of positive characteristic. (English) Zbl 1354.42054

Summary: Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply that the Gabor systems are Gabor frames is among the core problems in time-frequency analysis. In this paper, we give some simple and sufficient conditions that ensure a Gabor system \(\{M_{u(m)b}T_{u(n)a}g =: \chi_{m}(bx)g(x-u(n)a\}_{m,n\in\mathbb{N}_{0}}\) to be a frame for \(L^{2}(K)\). The conditions proposed are stated in terms of the Fourier transforms of the Gabor system’s generating functions.

MSC:

42C15 General harmonic expansions, frames
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
43A70 Analysis on specific locally compact and other abelian groups
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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References:

[1] R.J. Duffin and A.C. Shaeffer, A class of nonharmonic Fourier series, Transactions of the American Mathematical Society, vol. 72, pp. 341-366, 1952. · Zbl 0049.32401
[2] I. Daubechies, A. Grossmann and Y. Meyer, Painless non-orthogonal expansions, Journal of Mathematical Physics, vol. 27, no. 5, pp. 1271-1283, 1986. · Zbl 0608.46014
[3] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, Boston, 2015. · Zbl 1348.42033
[4] L. Debnath and F. A. Shah, Wavelet Transforms and Their Applications, Birkhäuser, New York, 2015.
[5] A. Ron and Z. Shen, Weyl-Heisenberg frames and Riesz bases in L2(Rd), Duke Mathematics Journal, vol. 89, pp. 237- 282, 1997. · Zbl 0892.42017
[6] K. Gröchenig, A.J. Janssen, N. Kaiblinger and GE. Pfander, Note on B-splines, wavelet scaling functions, and Gabor frames, IEEE Transactions and information Theory, vol. 49, no. 12, pp. 3318-3320, 2003. · Zbl 1286.94033
[7] P.G. Casazza and O. Christensen, Weyl-Heisenberg frames for subspaces of L2(R), Proceedings of American Mathematical Society, vol. 129, pp. 145-154, 2001. · Zbl 0987.42023
[8] K.Wang, Necessary and sufficient conditions for expansions of Gabor type, Analysis in Theory and Applications, vol. 22, pp. 155-171, 2006. · Zbl 1215.42050
[9] X.L. Shi and F. Chen, Necessary conditions for Gabor frames, Science in China : Series A. vol. 50, no. 2, pp. 276-284, 2007. · Zbl 1133.42046
[10] D. Li, G. Wu and X. Zhang, Two sufficient conditions in frequency domain for Gabor frames, Applied Mathematics Letters, vol. 24, pp. 506-511, 2011. · Zbl 1204.42054
[11] K. Gröchenig, Foundation of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
[12] H.G. Feichtinger and T. Strohmer, Advances in Gabor Analysis, Birkhäuser, Boston, 2003. · Zbl 1005.00015
[13] M.H. Taibleson, Fourier Analysis on Local Fields, Princeton University Press, Princeton, NJ, 1975. · Zbl 0319.42011
[14] D. Li and H.K. Jiang, Basic results Gabor frame on local fields, Chinese Annals of Mathematics: Series B, vol. 28, no. 2, pp. 165-176, 2007. · Zbl 1116.42010
[15] D. Li and M.A. Jun, Characterization of Gabor tight frames on local fields, Chinese Journal of Contemporary Mathematics, vol. 36, pp. 13-20, 2015. · Zbl 1340.42076
[16] F.A. Shah, Gabor frames on a half-line, Journal of Contemporary Mathematical Analysis, vol. 47, no. 5, pp. 251-260, 2012. · Zbl 1261.42054
[17] F.A. Shah and Abdullah, Wave packet frames on local fields of positive characteristic, Applied Mathematics and Computation, vol. 249, pp. 133-141, 2014. · Zbl 1338.42041
[18] F.A. Shah and Abdullah, A characterization of tight wavelet frames on local fields of positive characteristic, Journal of Contemporary Mathematical analysis, vol. 49, pp. 251-259, 2014. · Zbl 1332.42027
[19] F.A. Shah and M.Y. Bhat, Semi-orthogonal wavelet frames on local fields, Analysis, vol. 36, pp. 173-182, 2016. · Zbl 1346.42050
[20] D. Li, G. Wu and X. Yang, Unified conditions for wavelet frames, Georgian Mathematical Journal, vol. 18, pp. 761-776, 2011. · Zbl 1230.42040
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