论文标题
具有可分离时频移动的Gabor Riesz基地的表征
A characterization of Gabor Riesz bases with separable time-frequency shifts
论文作者
论文摘要
由窗口函数$ g \在l^2（\ mathbb {r}^d）$和可分开的集合$ g \和一个可分开的集合产生的gabor系统和一个可分离的集合$λ\ timesγ\ subset \ subset \ subset \ mathbb {r}^{2d} $是$ g $ g $ g $ \ natecal g（g，g $ g（ e^{2πiξ\ cdot t} g（t-x）\ right \} _ {（x，x，ξ）\在λ\ timesγ} $中。 Gabor分析中的基本问题之一是表征所有窗口和产生Gabor框架或Gabor正顺序基础的时频集。汉和王已经解决了由特征函数$ g =χ_Ω$生成的Gabor正顺基碱基的情况。在本文中，我们以这些结果为基础，并获得$ \ MATHCAL G（χ_Ω，λ\ timesγ）$的Riesz Gabor系统的完整表征。此外，对于某些类别的晶格$λ\ timesγ$，我们证明，多inving集的特征函数的必要条件可以用作Riesz Gabor基础的窗口函数，这是该集合必须是平铺设置。为了证明这一点，我们在Zak变换的零上开发了新的结果，并将这些结果连接到Gabor框架。
A Gabor system generated by a window function $g\in L^2(\mathbb{R}^d)$ and a separable set $Λ\times Γ\subset \mathbb{R}^{2d}$ is the collection of time-frequency shifts of $g$ given by $\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$. One of the fundamental problems in Gabor analysis is to characterize all windows and time-frequency sets that generate a Gabor frame or Gabor orthonormal basis. The case of Gabor orthonormal bases generated by characteristic functions $g=χ_Ω$ has been solved by Han and Wang. In this paper, we build on these results and obtain a full characterization of Riesz Gabor systems of the form $\mathcal G(χ_Ω, Λ\times Γ)$ when $Ω$ is a tiling of $\mathbb{R}^d$ with respect to $Λ$. Furthermore, for a certain class of lattices $Λ\times Γ$, we prove that a necessary condition for the characteristic function of a multi-tiling set to serve as a window function for a Riesz Gabor basis is that the set must be a tiling set. To prove this, we develop new results on the zeros of the Zak transform and connect these results to Gabor frames.
