A class of reproducing systems generated by a finite family in $$L^2({\mathbb {R}}_+)$$

Reproducing systems in $$L^2({\mathbb {R}})$$ such as wavelet and Gabor dual frames have been extensively studied, but reducing systems in $$L^2({\mathbb {R}}_+)$$ with $${{\mathbb {R}}_+}=(0,\,\infty )$$ have not. In practice, $$L^2({\mathbb {R}}_+)$$ models the causal space since the time variable cannot be negative. Due to $${\mathbb {R}}_+$$ not being a group under addition, $$L^2({\mathbb {R}}_+)$$ admits no nontrivial shift invariant system and thus admits no traditional wavelet or Gabor analysis. However, $$L^2({\mathbb {R}}_+)$$ admits nontrivial dilation systems due to $${\mathbb {R}}_+$$ being a group under multiplication. This paper addresses the frame theory of a class of dilation-and-modulation ( $${\mathcal{M}\mathcal{D}}$$ ) systems generated by a finite family in $$L^2({\mathbb {R}}_+)$$ . We obtain a parametric expression of $${\mathcal{M}\mathcal{D}}$$ -frames, and a density theorem for such $${\mathcal{M}\mathcal{D}}$$ -systems which is parallel to that of traditional Gabor systems in $$L^2({\mathbb {R}})$$ . It is well known that an arbitrary Gabor frame must admit dual frames with the same structure. Interestingly, it is not the case for $${\mathcal{M}\mathcal{D}}$$ -frames. We prove that an $${\mathcal{M}\mathcal{D}}$$ -frame admits $${\mathcal{M}\mathcal{D}}$$ -dual frames if and only if $$\log _ba$$ is an integer, where a and b are dilation and modulation parameters, respectively. And in this case, we characterize and express all $${\mathcal{M}\mathcal{D}}$$ -dual generators for an arbitrarily given $${\mathcal{M}\mathcal{D}}$$ -frame. Some examples are also provided.