发表机构
Faculty of Mathematics, University of Vienna（维也纳大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究确定第一埃尔米特函数的框架集，通过半正则Gabor框架刻画等方法证明Lyubarskii和Nes的猜想，得出次临界格点乘积的相关结论。
AI 中文摘要
我们确定了第一埃尔米特函数的框架集，证明了Lyubarskii和Nes的一个猜想。我们利用Gröchenig、Romero和Stöckler提出的半正则Gabor框架的刻画，将问题转化为关于存在一个非零高斯平移不变整函数F的问题，该函数具有有界系数，且其导数在间距为δ=ab的格点上消失。通过对F的N个平移取Wronskian行列式，我们将其临界点放大为重数至少为N-1的零点。重新标度后，该Wronskian仍是一个高斯平移不变函数。高斯零密度定理给出(N-1)/(Nδ)≤1。在构造允许的范围内改变N，会迫使所有使系统不成为框架的次临界格点乘积δ=ab等于某个整数q≥2对应的(q-1)/q。
英文摘要
We determine the frame set of the first Hermite function, thereby proving a conjecture of Lyubarskii and Nes. A characterization of semi-regular Gabor frames due to Gröchenig, Romero, and Stöckler reduces the problem to a uniqueness question for entire functions in a Gaussian shift-invariant space. We address this question by forming Wronskians of finitely many translates of a Gaussian shift-invariant function. Their common critical points become zeros of increasing multiplicity, while automorphy shows that the Wronskians remain within a Gaussian shift-invariant class. A sharp zero-density theorem then rules out all possible failures of the frame property except at the known rational obstructions. This yields the complete rectangular frame set of the first Hermite function and introduces Wronskian amplification as a new method in Gabor frame theory.
Comments17 pages; 41 references
