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

الأصل · EN

In the practice, time variable cannot be negative. The space L²(R+) of square integrable functions defined on the right half real line R+ models causal signal space. This paper focuses on a class of dilation-and-modulation systems in L²(R+). The density theorem for Gabor systems in L²(R) states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in L²(R) in terms of the index set alone-independently of window functions. The space L²(R+) admits no nontrivial Gabor system since R+ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in L²(R+) and the notion of Θ-transform matrix. Using Θ-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in L²(R+) under the condition that 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 L²(R+) is that ≤ 1. Simultaneously, we obtain a Θ-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in L²(R+).