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