Frame Sets and Zeros of Zak transforms of Extended Gaussians
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866915921188093952 |
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| author | Sun, Wenchang Zhou, Weiqi |
| author_facet | Sun, Wenchang Zhou, Weiqi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06641 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Frame Sets and Zeros of Zak transforms of Extended Gaussians Sun, Wenchang Zhou, Weiqi Classical Analysis and ODEs 42C15, 42C40 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. |
| title | Frame Sets and Zeros of Zak transforms of Extended Gaussians |
| topic | Classical Analysis and ODEs 42C15, 42C40 |
| url | https://arxiv.org/abs/2602.06641 |