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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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11271037). The authors thank the referees for their valuable comments.
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Li, Y., Zhang, W. Multi-window dilation-and-modulation frames on the half real line. Sci. China Math. 63, 2423–2438 (2020). https://doi.org/10.1007/s11425-018-9468-8
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DOI: https://doi.org/10.1007/s11425-018-9468-8
Keywords
- frame
- wavelet frame
- Gabor frame
- dilation-and-modulation frame
- multi-window dilation-and-modulation frame