Multi-window Wilson systems in ℓ 2 (ℤ) for general lattices with rational density

In this paper, we introduce the definition of multi-window Wilson systems in ℓ 2 (ℤ) for a class of general lattices with rational density, and obtain a necessary and sufficient condition for two multi-window Wilson Bessel sequences being dual frames for ℓ 2 (ℤ). Furthermore, we provide characterizations of a multi-window Wilson system to be a tight frame (resp. an orthonormal basis) for ℓ 2 (ℤ). When the window functions satisfy certain conditions, we show that a Wilson frame for ℓ 2 (ℤ) can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame. Several examples of (multi-window) Wilson tight frames are presented to illustrate the theoretical results. The results of this paper are new even in the single-window case and for rectangular lattices.