Document Type: Research Paper

Author

Department of Mathematics, Vali-e-Asr University of Rafsanjan, P.O.Box 546, Rafsanjan, Iran.

Abstract

The duals of Gabor frames have an essential role in reconstruction of signals. In this paper we find a necessary and sufficient  condition for two Gabor systems $\left(\chi_{\left[c_1,d_1\right)},a,b\right)$ and $\left(\chi_{\left[c_2,d_2\right)},a,b\right)$ to form dual frames for $L_2\left(\mathbb{R}\right)$, where $a$ and $b$ are positive numbers and $c_1,c_2,d_1$ and $d_2$ are real numbers such that $c_1<d_1$ and $c_2<d_2$.

Keywords

[1] A.A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35 (2013), pp. 535-540.

[2] A.A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iranian Math. Soc., 42(6), (2016), pp. 1305 -1314.

[3] A. Askari Hemmat, A. Safapour and Z. Yazdani Fard, Coherent Frames, Sahand Commun. Math. Anal., 11(1) (2018), pp. 1-11.

[4] J.J. Benedetto, Frame decomposition, sampling and uncertainty principle inequalities in wavelets, Mathematics and applications (Eds. J.J. Benedetto and M. W. Frazier.), CRC Press., Boca Raton, FL, 1994 , Chapt. 7.

[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser., Boston, Basel, Berlin, 2002.

[6] R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[7] H.G. Feichtinger and T. Strohmer, Eds, Gabor Analysis and Algorithms-Theory and Applications, Birkhauser., Boston, 1998.

[8] A. Ghaani Farashahi, Continuous partial Gabor transform for semidirect product of locally compact groups, Bull. Malays. Math. Sci. Soc., 38(2) (2015), pp. 779-803.

[9] A. Ghaani Farashahi and R.A. Kamyabi-Gol, Continuous Gabor transform for a class of non-Abelian groups, B. Belg. Math. Soc-Sim., 19(4) (2012), pp. 683-701.

[10] F. Ghobadzadeh and A. Najati, G-dual Frames in Hilbert $C^*$-module Spaces, Sahand Commun. Math. Anal., 11(1) (2018), pp. 65-79.

[11] K. Grochenig, Aspects of Gabor analysis on locally compact Abelian groups, Gabor analysis and Algorithms, ANHA, Birkhauser., Boston MA, 1998, 211-231.

[12] C. Heil and D. Walnut, Continuous and discrete wavelet transform, SIAM Rev., 31 (1969), pp. 628-666.

[13] M. Mirzaee Azandaryani, Approximate Duals of g-frames and Fusion Frames in Hilbert $C^*$-modules, Sahand Commun. Math. Anal., 15(1) (2019), pp. 135-146.

[14] A.J.E.M. Janssen, The duality condition for Weyl-Heisenberg frames, In Gabor analysis: theory and application (Eds. H.G. Feichtinger and T. Strhmer). Birkhauser., Boston, 1998.

[15] A.J.E.M. Janssen, Zak transforms with few zeros and the tie, In: Advances in Gabor Analysis (Eds.: H.G. Feichtinger and T. Strohmer), Birkhauser., Boston, 2003.

[16] M. Rashidi-Kouchi, Frames in super Hilbert modules, Sahand Commun. Math. Anal., 09(1) (2018), pp. 129-142.

[17] A. Ron and Z. Shen, Weyl-Heisenberg systems and Riesz bases in $L_2(mathbb{R^d)$, Duke. Math. J., 89 (1997), pp. 237-282.

[18] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press., New York, 1980.