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

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 $\Theta_{a}$-transform in $L^{2}(\Bbb R_{+})$, using $\Theta_{a}$-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.