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 spaceL(2)(Double-struck capital R+).L-2(Double-struck capital R+) can be identified as a subspace ofL(2)(Double-struck capital R), which consists of allL(2)(Double-struck capital R)-functions supported on Double-struck capital R(+)but not closed under the Fourier transform. Therefore, the Fourier transform method does not work inL(2)(Double-struck capital R+). Herein, we introduce the notion of Theta(a)-transform inL(2)(Double-struck capital R+) and characterize the dilation-and-modulation frames and dual frames inL(2)(Double-struck capital R+) using the Theta(a)-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame forL(2)(Double-struck capital 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.