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arXiv 1708.05941math.FA
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半实线上的多窗口伸缩调制帧

Multi-window dilation-and-modulation frames on the half real line

Yun-Zhang Li, Wei Zhang

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摘要：

小波系统和Gabor系统分别基于伸缩和调制算子。它们已被广泛研究。然而，伸缩和调制系统没有，它们不能从小波或Gabor系统导出。本文研究了因果信号空间$L^{2}(\Bbb R_{+})$中的一类伸缩调制系统。$L^{2}(\Bbb R_{+})$可以被识别为由$\Bbb R_{+}$上支持的所有$L^{2}(\Bbb R)$函数组成的$L^{2}(\Bbb R)$的子空间，并且在傅里叶变换下是不封闭的。因此，傅里叶变换方法在$L^{2}(\Bbb R_{+})$中不起作用。本文在$L^{2}(\Bbb R_{+})$中引入了$Θ_{a}$-变换的概念，利用$Θ_{a}$-变换刻画了$L^{2}(\Bbb R_{+})$中的伸缩调制帧和对偶帧，并给出了$L^{2}(\Bbb R_{+})$中一般伸缩调制帧的所有具有相同结构的子帧的显式表达式.有趣的是，我们证明了这种形式的任意帧总是非冗余的，只要生成器的数量是$1$，总是冗余的，只要它是大于$1$。最后通过一些例子说明了所得结果的一般性.

英文摘要：

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively. They have been extensively studied. However, dilation-and-modulation systems have not, and they cannot be derived from wavelet or Gabor systems. In this paper, we investigate a class of dilation-and-modulation systems in the causal signal space $L^{2}(\Bbb R_{+})$. $L^{2}(\Bbb R_{+})$ can be identified a subspace of $L^{2}(\Bbb R)$ consisting of all $L^{2}(\Bbb R)$-functions supported on $\Bbb R_{+}$, and is unclosed under the Fourier transform. So the Fourier transform method does not work in $L^{2}(\Bbb R_{+})$. In this paper, we introduce the notion of $Θ_{a}$-transform in $L^{2}(\Bbb R_{+})$, using $Θ_{a}$-transform we characterize dilation-and-modulation frames and dual frames in $L^{2}(\Bbb R_{+})$; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for $L^{2}(\Bbb R_{+})$. Interestingly, we prove that an arbitrary frame of this form is always nonredundant whenever the number of the generators is $1$, and is always redundant whenever it is greater than $1$. Some examples are also provided to illustrate the generality of our results.

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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