Linear independence of a finite set of time-frequency shifts

Abstract

This paper introduces an open conjecture in time-frequency analysis on the linear independence of a finite set of time-frequency shifts of a given L² function. Firstly, background and motivation for the conjecture are provided. Secondly, the main progress of this linear independence in the past twenty five years is reviewed. Finally, the partial results of the high dimensional case and other cases for the conjecture are briefly presented.

Keywords

Time-frequency shift / Gabor frame / wavelet / linear independence

E-mail: [e-mail]

References

[1] AntezanaJBrunaJPujalsE. Linear independence of time frequency translates in L^p spaces. J Fourier Anal Appl, 2020, 26(4): 63 (15 pp)

[2] Balan R. The noncommutative Wiener lemma, linear independence, and spectral properties of the algebra of time-frequency shift operators. Trans Ams Math Soc 2008; 360(7): 3921–3941

[3] Balan R, Krishtal I. An almost periodic noncommutative Wiener’s lemma. J Math Anal Appl 2010; 370(2): 339–349

[4] Benedetto J J, Bourouihiya A. Linear independence of finite Gabor systems determined by behavior at infinity. J Geom Anal 2015; 25(1): 226–254

[5] Bownik M, Speegle D. Linear independence of Parseval wavelets. Illinois J Math 2010; 54(2): 771–785

[6] Bownik M, Speegle D. Linear independence of time-frequency translates of functions with faster than exponential decay. Bull Lond Math Soc 2013; 45(3): 554–556

[7] Bownik M, Speegle D. Linear independence of time-frequency translates in Rd. J Geom Anal 2016; 26(3): 1678–1692

[8] ChristensenO. An Introduction to Frames and Riesz Bases. 2nd Ed, Applied and Numerical Harmonic Analysis, MA: Birkhäuser, 2016.

[9] Christensen O, Hasannasab M. Gabor frames in l2(Z) and linear dependence. J Fourier Anal Appl 2019; 25(1): 101–107

[10] Christensen O, Lindner A M. Lower bounds for finite wavelet and Gabor systems. Approx Theory Appl (N S) 2001; 17(1): 18–29

[11] CurreyBOussaV. HRT conjecture and linear independence of translates on the Heisenberg group. 2018

[12] DaubechiesI, Ten Lectures on Wavelets. CBMS-NSF Regional Conference Series in Applied Mathematics, Vol 61. Philadelphia, PA: SIAM, 1992.

[13] Demeter C. Linear independence of time-frequency translates for special configurations. Math Res Lett 2010; 17(4): 761–779

[14] Demeter C, Gautam S Z. On the finite linear independence of lattice Gabor systems. Proc Am Math Soc 2013; 141(5): 1735–1747

[15] Demeter C, Zaharescu A. Proof of the HRT conjecture for (2, 2) configurations. J Math Anal Appl 2012; 388(1): 151–159

[16] Gröchenig K. Linear independence of time-frequency shifts?. Monatsh Math 2015; 177(1): 67–77

[17] HeilC. Linear independence of finite Gabor systems. In: Harmonic Analysis and Applications, Appl Numer Harmon Anal, Boston, MA: Birkhäuser Boston, 2006, 171–206

[18] Heil C, Ramanathan J, Topiwala P. Linear independence of time-frequency translates. Proc Amer Math Soc 1996; 124(9): 2787–2795

[19] HeilCSpeegleD. The HRT conjecture and the Zero Divisor Conjecture for the Heisenberg group. In: Excursions in Harmonic Analysis, Appl and Numer Harmon Anal, Vol 3. Cham: Birkhäuser, 2015, 159–176

[20] Jitomirskaya S Y. Metal-insulator transition for the almost Mathieu operator. Ann Math(2) 1999; 150(3): 1159–1175

[21] Kreisel M. Linear independence of time frequency shifts up to extreme dilations. J Fourier Anal Appl 2019; 25(6): 3214–3219

[22] Kutyniok G. Linear independence of time-frequency shifts under a generalized Schrödinger representation. Arch Math (Basel) 2002; 78(2): 135–144

[23] Lawrence J, Pfander G E, Walnut D. Linear independence of Gabor systems in finite dimensional vector spaces. J Fourier Anal Appl 2005; 11(6): 715–726

[24] LiD F. Mathematical Theory of Wavelet Analysis. Beijing: Science Press, 2017(in Chinese)

[25] Linnell P A. Von Neumann algebras and linear independence of translates. Proc Amer Math Soc 1999; 127(11): 3269–3277

[26] Liu W C. Letter to the Editor: Proof of the HRT conjecture for almost every (1, 3) configuration. J Fourier Anal Appl 2019; 25(4): 1350–1360

[27] NicolaFTrapassoS I. A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform. J Fourier Anal Appl, 2020, 26(4): 68 (7 pp)

[28] Okoudjou K A. Extension and restriction principles for the HRT conjecture. J Fourier Anal Appl 2019; 25(4): 1874–1901

[29] OussaV. New insights into the HRT conjecture. 2019

[30] Rosenblatt J M. Linear independence of translations. J Aust Math Soc Ser A 1995; 59(1): 131–133

[31] Rosenblatt J. Linear independence of translations. Int J Pure Appl Math 2008; 45(2): 463–473

[32] Saliani S. l2-linear independence for the system of integer translates of a square integrable function. Proc Amer Math Soc 2013; 141(3): 937–941

[33] Saliani S. lp-linear independence for the system of integer translates. J Fourier Anal 2014; 20(4): 766–783

[34] StroockD W. Remarks on the HRT conjecture. In: Memoriam Marc Yor–Sćéminar on Probabilités Theory XLVII, Lecture Notes in Math, Vol 2137. Cham: Springer, 2015, 603–617

Acknowledgements

This work was supported in part by the National Natural Science Foundation of China (Grant No. 61471410) and the Construction Fund for Subject Innovation Term of Wuhan Textile University (No. 201401023).

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