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, 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 Θ_a-transform in L²(ℝ₊) and characterize the dilation-and-modulation frames and dual frames in L²(ℝ₊) using the Θ_a-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.