We show that every rationally sampled dilation-and-modulation system is unitarily equivalent to a multi-window Gabor system with special structures. As a consequence, the study of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ reduces to that of multi-window Gabor systems in $$L^2({\mathbb {R}})$$ with special structures.

中文翻译：

---

关于 $${L^2({\pmb {\mathbb {R}}_+)}}$$ 的膨胀和调制帧的注释

我们表明，每个合理采样的扩张和调制系统均等价于具有特殊结构的多窗口 Gabor 系统。因此，$L^2({\mathbb {R}}_+)$$ 中的扩张和调制系统的研究减少到 $$L^2({\ mathbb {R}})$$ 具有特殊结构。