arXiv 2202.063432022-02-13math.FA

具有可分离时频移的Gabor Riesz基的一个特征

A characterization of Gabor Riesz bases with separable time-frequency shifts

Christina Frederick, Azita Mayeli

摘要：

由窗函数$g\in L^2(\mathbb{R}^d)$和可分集$Λ\times Γ\subset \mathbb{R}^{2d}$生成的Gabor系统是由$\mathcal G(g, Λ\times Γ) = \left\{ e^{2πi ξ\cdot t}g(t-x)\right\}_{ (x,ξ)\in Λ\times Γ}$给出的$g$的时频偏移的集合。Gabor分析中的一个基本问题是刻画所有生成Gabor框架或Gabor正交基的窗口和时频集。由特征函数$g=χ_Ω$生成的Gabor正交基的情形已由Han和Wang解决。在本文中，我们建立在这些结果，并获得充分的表征Riesz Gabor系统的形式$\mathcal G(χ_Ω, Λ\times Γ)$时，$Ω$是一个平铺的$\mathbb{R}^d$关于$Λ$。进一步，对于一类格$Λ\times Γ$，证明了多块集的特征函数作为Riesz-Gabor基的窗函数的必要条件是该集必须是块集.为了证明这一点，我们开发的Zak变换的零点的新结果，并将这些结果的Gabor帧。

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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