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 $L^2(\Bbb R_+)$ of square integrable functions defined on the right half real line $\Bbb R_+$ models causal signal space. This paper focuses on a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$. The density theorem for Gabor systems in $L^2(\Bbb R)$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $L^2(\Bbb R)$ in terms of the index set alone-independently of window functions. The space $L^2(\Bbb R_+)$ admits no nontrivial Gabor system since $\Bbb R_+$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $L^2(\Bbb R_+)$ and the notion of $\Theta$-transform matrix. Using $\Theta$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $L^2(\Bbb R_+)$ under the condition that $\log_ba$ 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 $L^2(\Bbb R_+)$ is that $\log_ba \leq 1$. Simultaneously, we obtain a $\Theta$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $L^2(\Bbb R_+)$.