2017/08/20 by Yun‐Zhang Li, Wei Zhang, Li, Yun-Zhang +1
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.1708.05941
openalex publication_date 2017/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 L2(\Bbb R+). L2(\Bbb R+) can be identified a subspace of L2(\Bbb R) consisting of all L2(\Bbb R)-functions supported on \Bbb R+, and is unclosed under the Fourier transform. So the Fourier transform method does not work in L2(\Bbb R+). In this paper, we introduce the notion of Θa-transform in L2(\Bbb R+), using Θa-transform we characterize dilation-and-modulation frames and dual frames in L2(\Bbb R+); and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L2(\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.