Geodesic
On the $abc$-problem in Weyl-Heisenberg frames
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 2, pp. 447-458 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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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     title = {On the $abc$-problem in {Weyl-Heisenberg} frames},
     journal = {Czechoslovak Mathematical Journal},
     pages = {447--458},
     year = {2014},
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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

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