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arXiv 0903.4989math.FAmath.CA
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海森堡群的Gabor场和小波集

Gabor fields and wavelet sets for the Heisenberg group

Bradley Currey, Azita Mayeli

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摘要：

我们研究了$\Bbb R^2$上的单生成小波系统，这些系统与海森堡群$N$上的秩一小波系统自然相关。我们证明了生成元上的一个必要条件，使得任何这样的系统都是Parseval框架.给定$N$的对偶的一个合适的子集$I$，我们给出了与$I$相关的Parseval框架小波的一个显式构造.我们称$g\in L^2(I\times \Bbb R)$是$I$上的Gabor域，如果对a.e. $λ\in I$，$|λ|^{1/2} g(λ,\cdot)$是$L^2(\Bbb R)$的Parseval框架的Gabor生成器，并且如果$I$上的每个Gabor域都是$L^2(\Bbb R^2)$的Parseval框架（母）小波，则$I$是海森堡小波集。证明了$I$是Heisenberg小波集的充要条件是$I$既与单位区间的子集平移同余又与Shannon集伸缩同余.

英文摘要：

We study singly-generated wavelet systems on $\Bbb R^2$ that are naturally associated with rank-one wavelet systems on the Heisenberg group $N$. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an explicit construction for Parseval frame wavelets that are associated with $I$. We say that $g\in L^2(I\times \Bbb R)$ is Gabor field over $I$ if, for a.e. $λ\in I$, $|λ|^{1/2} g(λ,\cdot)$ is the Gabor generator of a Parseval frame for $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if every Gabor field over $I$ is a Parseval frame (mother-)wavelet for $L^2(\Bbb R^2)$. We then show that $I$ is a Heisenberg wavelet set if and only if $I$ is both translation congruent with a subset of the unit interval and dilation congruent with the Shannon set.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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