On the $abc$-problem in Weyl-Heisenberg frames

Title:

On the $abc$-problem in Weyl-Heisenberg frames

Creator:

He, Xinggang and Li, Haixiong

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:b526eae4-b4cb-4cc0-b7bd-efc8362d04af
uuid:b526eae4-b4cb-4cc0-b7bd-efc8362d04af

Subject:

$abc$-problem, Weyl-Heisenberg frame, and Zak transform

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

447-458

Source:

Czechoslovak Mathematical Journal | 2014 Volume:64 | Number:2

Harvested from:

CDK

Metadata only:

false