Korean J. Math. Vol. 28 No. 4 (2020) pp.877-888
DOI: https://doi.org/10.11568/kjm.2020.28.4.877

Gabor frames in $l^2 (\mathbb Z)$ from Gabor frames in $L^2 (\mathbb R)$

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Published: Dec 30, 2020
Keywords:
Hilbert space, Gabor frame, Orthonormal basis, Unitary operator, Window sequence.
Mathematics Subject Classification:
42C15, 47B90, 94A12

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Jineesh Thomas
Research department of Mathematics St.Thomas College Palai Kottayam, Kerala, India, 686574.
Madhavan Namboothiri N M
Department of Mathematics Government College Kottayam Nattakom P O, Kerala, India, 686013.
Eldo Varghese
Research department of Mathematics St.Thomas College Palai Kottayam, Kerala, India, 686574.

Abstract

In this paper we discuss about the image of Gabor frame under a unitary operator and derive a sufficient condition under which a unitary operator from $L^2 (\mathbb R)$ to $l^2 (\mathbb Z)$ maps Gabor frame in $L^2 (\mathbb R)$ to a Gabor frame in $l^2 (\mathbb Z)$.


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References

[1] Z. Amiri, M. A. Dehghan, E. Rahimib and L. Soltania, Bessel subfusion sequences and subfusion frames, Iran. J. Math. Sci. Inform. 8 (1) (2013), 31–38. Google Scholar

[2] P. G. Cazassa and G. Kutyniok, Frames of subspaces. Wavelets, frames and operator theory, Contemp. Math., Vol. 345, Amer. Math. Soc., Providence, R. I. (2004), 87–113. Google Scholar

[3] P. G. Cazassa and G. Kutyniok, Finite Frames Theory and Applications, Applied and Numerical Harmonic Analysis series, Birkh ̈auser, (2013). Google Scholar

[4] O. Christensen, An Introduction to Frames and Riesz Bases, Second Edition, Birkh ̈auser, 2016. Google Scholar

[5] O. Christensen and Y. C. Eldar, Oblique dual frames and shift-invariant spaces, Appl.Comput. Harmon. Anal. 17 (2004), 48–68. Google Scholar

[6] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions , J. Math. Physics 27 (1986) 1271–1283. Google Scholar

[7] R. J. Duffin and A. C.Schaeffer, A class of non-harmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. Google Scholar

[8] T.C.Easwaran Nambudiri, K. Parthasarathy, Characterization of Weyl- Heisenberg frame operators, Bull.Sci. Math. 137 (2013) 322–324. Google Scholar

[9] T.C.Easwaran Nambudiri, K. Parthasarathy, Generalised Weyl-Heisenberg frame operators, Bull.Sci. Math. 136 (2012) 44–53. Google Scholar

[10] Feichtinger H.G., Strohmer T., Gabor Analysis and Algorithms: Theory and Applications, Birkh ̈auser, Boston, 1998. Google Scholar

[11] D.Gabor, Theory of communication, J.IEEE. 93 (1946) 429–457. Google Scholar

[12] Gotz E.Pfander, Gabor frames in finite dimensions, Birkh ̈auser, (2010). Google Scholar

[13] K.Gr ̈ochenig, Foundations of Time Frequency Analysis, Birkh ̈auser, 2001. Google Scholar

[14] A. J. E. M.Janssen, Gabor representation of generalized functions, J. Math.Anal. Appl. 83 (1981) 377–394. Google Scholar

[15] A. J. E. M.Janssen, From continuous to discrete Weyl-Heisenberg framesthrough sampling, J. Four. Anal. Appl. 3 (5) (1971) 583–596. Google Scholar

[16] S. Li and H. Ogawa,Pseudoframes for subspaces with applications, J. FourierAnal.Appl. 10 (2004), 409–431. Google Scholar