框架集与扩展高斯函数的Zak变换零点
Frame Sets and Zeros of Zak Transforms of Extended Gaussians
AI总结：
研究扩展高斯函数的框架集及其Zak变换零点，证明其框架集最大且Zak变换在单位正方形内有唯一简单零点，扩展了传统高斯函数的结果。
摘要：
设a,b,c∈C且Re(a)<0，证明扩展高斯函数e^{ax²+bx+c}具有最大框架集（即其框架集恰好由所有正对(α,β)组成，其中αβ<1），且其Zak变换在单位正方形[0,1)²内有唯一简单零点（特别是当b=0时，零点位于单位正方形中心）。这些结果扩展了传统高斯函数（当a<0且b,c∈R时）的相同结论，并补充了观察到的结论：如果连续Wiener函数具有最大框架集，则其Zak变换在单位正方形内有唯一简单零点。框架集最大性的证明结合了元摄动表示与标准高斯的古典密度结果。零点唯一性的证明依赖于theta函数的性质。
英文摘要：
Let $a,b,c\in\mathbb C$ with $\re(a)<0$, we show that the extended Gaussian $e^{ax^2+bx+c}$ has maximal frame set (i.e., its frame set consists of precisely all positive pairs $(α,β)$ with $αβ<1$), and its Zak transform has a unique simple zero in the unit square $[0,1)^2$ (in particular, the zero is at the center of the unit square if $b=0$). These statements extend the same results of the usual Gaussian (the cases when $a<0$ and $b,c\in\mathbb R$), and add more instances to the observation that if a continuous Wiener function has maximal frame set, then its Zak transform has a unique simple zero in the unit square. The proof of the maximality of the frame set combines metaplectic representation with a classical density result of the standard Gaussian. The proof of the uniqueness of the zero relies on properties of the theta function.
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。
