Multi-window dilation-and-modulation frames on the half real line |
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Authors: | Li Yunzhang Zhang Wei |
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Institution: | 1.College of Applied Sciences, Beijing University of Technology, Beijing, 100124, China ;2.School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou, 450046, China ; |
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Abstract: | Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively, and have been studied extensively. However, dilation-and-modulation systems cannot be derived from wavelet or Gabor systems. This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L2(ℝ+). L2(ℝ+) can be identified as a subspace of L2(ℝ), which consists of all L2(ℝ)-functions supported on ℝ+ but not closed under the Fourier transform. Therefore, the Fourier transform method does not work in L2(ℝ+). Herein, we introduce the notion of Θa-transform in L2(ℝ+) and characterize the dilation-and-modulation frames and dual frames in L2(ℝ+) using the Θa-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L2(ℝ+). Furthermore, it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1. Finally, some examples are provided to illustrate the generality of our results. |
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