| Title:
|
On the $abc$-problem in Weyl-Heisenberg frames (English) |
| Author:
|
He, Xinggang |
| Author:
|
Li, Haixiong |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
2 |
| Year:
|
2014 |
| Pages:
|
447-458 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
$abc$-problem |
| Keyword:
|
Weyl-Heisenberg frame |
| Keyword:
|
Zak transform |
| MSC:
|
42C15 |
| MSC:
|
42C40 |
| idZBL:
|
Zbl 06391504 |
| idMR:
|
MR3277746 |
| DOI:
|
10.1007/s10587-014-0111-z |
| . |
| Date available:
|
2014-11-10T09:45:31Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144008 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[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). Zbl 1027.42025, MR 1955931 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| . |
