AI 中文总结
本文针对极点对称的有理窗函数$g_n$，利用Zak变换建立Gabor框架的有限维判据，得出三类精确密度下的新框架结果，并证明$g_1$生成的Gabor系统在特定有理密度区间内为框架。
AI 中文摘要
我们研究有理窗函数族的Gabor框架问题，该族窗函数为$g_n(x)=\frac{\left(\prod_{k=1}^{\lfloor n/2\rfloor}(x^2-k^2)\right)\cdot\begin{cases}1, & n \text{ 为偶数},\x, & n \text{ 为奇数},\end{cases}}{\prod_{k=1}^{n+1}(x^2+k^2)},\quad n\in \mathbb{N}$，其极点在虚轴上呈对称对出现。这些窗函数不属于现有有理Herglotz函数及指数多项式比值的框架集刻画所覆盖的类别。对于有理密度（$\alpha\beta = p/q$），我们利用Zak变换将Gabor框架条件直接映射为有限维等价判据，该判据完全由相关多项式矩阵的秩决定。应用该框架，我们建立了精确密度$\alpha\beta = 1/(2n+1)$（对应$g_n$，$n \ge 1$）、$\alpha\beta = 1/(n+1)$（对应$g_n$，$n \ge 2$）及$\alpha\beta=\frac{2N+1}{2N+3}$，$N\ge 1$（对应$g_1$）的新框架结果。此外，我们证明由$g_1$生成的Gabor系统对每个满足$\alpha\beta=\frac{p}{q}\in (1/2,2/3)\cup (2/3,5/7)$的有理密度都构成一个框架。
英文摘要
We investigate the Gabor frame problem for a rational window $$g(x)= \frac{x}{(x^2+1)(x^2+4)}.$$ This window falls outside the classes covered by existing frame set characterizations for rational Herglotz functions and ratios of exponential polynomials. For rational densities ($αβ= p/q$), we leverage the Zak transform to map the Gabor frame condition directly to a finite-dimensional equivalence criterion, which is completely determined by the rank of an associated polynomial matrix. Applying this framework, we establish new frame results for every rational density $$αβ=\frac{p}{q}\in \left(0,\frac{1}{2}\right)\cup\left(\frac{1}{2},\frac{2}{3}\right)\cup \left(\frac{2}{3},\frac{5}{7}\right].$$
Comments32 pages, 3 figures
