The density theorem of a class of dilation-and-modulation systems on the half real line

In the practice, time variable cannot be negative. The space $L^2({\mathbb{R}}_{+})$ of square integrable functions defined on the right half real line ${\mathbb{R}}_{+}$ models causal signal space. This paper focuses on a class of dilation-and-modulation systems in $L^2({\mathbb{R}}_{+})$. The density theorem for Gabor systems in $L^2(\mathbb{R})$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $L^2(\mathbb{R})$ in terms of the index set alone-independently of window functions. The space $L^2({\mathbb{R}}_{+})$ admits no nontrivial Gabor system since ${\mathbb{R}}_{+}$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $L^2({\mathbb{R}}_{+})$ and the notion of $\mathrm{\Theta }$-transform matrix. Using $\mathrm{\Theta }$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $L^2({\mathbb{R}}_{+})$ under the condition that ${\log }_{b}a$ is a positive rational number, where $a$ and $b$ are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $L^2({\mathbb{R}}_{+})$ is that ${\log }_{b}a\le 1$. Simultaneously, we obtain a $\mathrm{\Theta }$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $L^2({\mathbb{R}}_{+})$.