Geodesic
On the $abc$-problem in Weyl-Heisenberg frames
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 447-458
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Let $a,b,c>0$. We investigate the characterization problem which asks for a classification of all the triples $(a,b,c)$ such that the Weyl-Heisenberg system $\{{\rm e}^{2\pi {\rm i}mbx} \* \chi _{[na,na+c)}\colon m,n\in {\mathbb Z}\}$ is a frame for $L^2({\mathbb R})$. It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b=1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of $a$ being a rational number. We prove some special cases of values for $c$ and $a$ that do not produce a frame, which expands earlier works.
Let $a,b,c>0$. We investigate the characterization problem which asks for a classification of all the triples $(a,b,c)$ such that the Weyl-Heisenberg system $\{{\rm e}^{2\pi {\rm i}mbx} \* \chi _{[na,na+c)}\colon m,n\in {\mathbb Z}\}$ is a frame for $L^2({\mathbb R})$. It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where $b=1$ and only let $a$ and $c$ vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of $a$ being a rational number. We prove some special cases of values for $c$ and $a$ that do not produce a frame, which expands earlier works.
DOI : 10.1007/s10587-014-0111-z
Classification : 42C15, 42C40
Keywords: $abc$-problem; Weyl-Heisenberg frame; Zak transform 
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He, Xinggang; Li, Haixiong. On the $abc$-problem in Weyl-Heisenberg frames. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 447-458. doi: 10.1007/s10587-014-0111-z

[1] Casazza, P. G.: Modern tools for Weyl-Heisenberg (Gabor) frame theory. Adv. Imag. Elec. Phys. 115 (2000), 1-127.

[2] Casazza, P. G., Kalton, N. J.: Roots of complex polynomials and Weyl-Heisenberg frame sets. Proc. Am. Math. Soc. 130 (2002), 2313-2318. | DOI | MR | Zbl

[3] Gröchenig, K., Stöckler, J.: Gabor frames and totally positive functions. Duke Math. J. 162 (2013), 1003-1031. | DOI | MR | Zbl

[4] Gu, Q., Han, D.: When a characteristic function generates a Gabor frame. Appl. Comput. Harmon. Anal. 24 (2008), 290-309. | DOI | MR | Zbl

[5] Heil, C.: History and evolution of the density theorem for Gabor frames. J. Fourier Anal. Appl. 13 (2007), 113-166. | DOI | MR | Zbl

[6] Janssen, A. J. E. M.: Some Weyl-Heisenberg frame bound calculations. Indag. Math., New Ser. 7 (1996), 165-183. | DOI | MR | Zbl

[7] Janssen, A. J. E. M.: Zak transforms with few zeros and the tie. Advances in Gabor Analysis H. G. Feichtinger et al. Applied and Numerical Harmonic Analysis Birkhäuser, Basel 31-70 (2003). | MR | Zbl

[8] Janssen, A. J. E. M., Strohmer, T.: Hyperbolic secants yield Gabor frames. Appl. Comput. Harmon. Anal. 12 (2002), 259-267. | DOI | MR | Zbl

[9] Lyubarskij, Y. I.: Frames in the Bargmann space of entire functions. Entire and Subharmonic Functions Advances in Soviet Mathematics 11 American Mathematical Society, Providence (1992), 167-180. | MR | Zbl

[10] Seip, K.: Density theorems for sampling and interpolation in the Bargmann-Fock space I. J. Reine Angew. Math. 429 (1992), 91-106. | MR | Zbl

[11] Seip, K., Wallstén, R.: Density theorems for sampling and interpolation in the Bargmann-Fock space II. J. Reine Angew. Math. 429 (1992), 107-113. | MR | Zbl

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