Beurling density theorems for sampling and interpolation on the flat cylinder
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| Format: | Preprint |
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2024
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| author | Abreu, Luis Daniel Luef, Franz Ziyat, Mohammed |
| author_facet | Abreu, Luis Daniel Luef, Franz Ziyat, Mohammed |
| contents | We consider the Fock space weighted by $e^{-α|z|^{2}}$, of entire and quasi-periodic (modulo a weight dependent on $ν$) functions on ${C}$. The quotient space $\mathbb{C}/\mathbb{Z}$, called `The flat cylinder', is represented by the vertical strip $[0,1)\times \mathbb{R}$, which tiles ${C}$ by ${Z}$-translations and is therefore a fundamental domain for $\mathbb{C}/\mathbb{Z}$. Our main result gives a complete characterization of the sets $Z\subset Λ\left( \mathbb{Z}\right) $ that are sets of sampling or interpolation, in terms of concepts of upper and lower Beurling densities, $ D^{+}(Z)$ and $D^{-}(Z)$, adapted to the geometry of $\mathbb{C}/\mathbb{Z}$. The critical `Nyquist density' is the real number $\frac{α}{π}$, meaning that the condition $D^{-}(Z)>\frac{α}{π}$ characterizes sets of sampling, while the condition $D^{+}(Z)<\frac{α}{π}$ characterizes sets of interpolation. The results can be reframed as a complete characterization of Gabor frames and Riesz basic sequences (given by arbitrary discrete sets in $Z\subset Λ\left( \mathbb{Z}\right) $), with time-periodized Gaussian windows (theta-Gaussian), for spaces of functions $f$, measurable in $\mathbb{R}$, square-integrable in $(0,1)$, and quasi-periodic with respect to integer translations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_21094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Beurling density theorems for sampling and interpolation on the flat cylinder Abreu, Luis Daniel Luef, Franz Ziyat, Mohammed Functional Analysis Classical Analysis and ODEs Complex Variables We consider the Fock space weighted by $e^{-α|z|^{2}}$, of entire and quasi-periodic (modulo a weight dependent on $ν$) functions on ${C}$. The quotient space $\mathbb{C}/\mathbb{Z}$, called `The flat cylinder', is represented by the vertical strip $[0,1)\times \mathbb{R}$, which tiles ${C}$ by ${Z}$-translations and is therefore a fundamental domain for $\mathbb{C}/\mathbb{Z}$. Our main result gives a complete characterization of the sets $Z\subset Λ\left( \mathbb{Z}\right) $ that are sets of sampling or interpolation, in terms of concepts of upper and lower Beurling densities, $ D^{+}(Z)$ and $D^{-}(Z)$, adapted to the geometry of $\mathbb{C}/\mathbb{Z}$. The critical `Nyquist density' is the real number $\frac{α}{π}$, meaning that the condition $D^{-}(Z)>\frac{α}{π}$ characterizes sets of sampling, while the condition $D^{+}(Z)<\frac{α}{π}$ characterizes sets of interpolation. The results can be reframed as a complete characterization of Gabor frames and Riesz basic sequences (given by arbitrary discrete sets in $Z\subset Λ\left( \mathbb{Z}\right) $), with time-periodized Gaussian windows (theta-Gaussian), for spaces of functions $f$, measurable in $\mathbb{R}$, square-integrable in $(0,1)$, and quasi-periodic with respect to integer translations. |
| title | Beurling density theorems for sampling and interpolation on the flat cylinder |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables |
| url | https://arxiv.org/abs/2412.21094 |