A discrete Hardy uncertainty principle
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909014483271680 |
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| author | Lysen, Torgeir Keun |
| author_facet | Lysen, Torgeir Keun |
| contents | We show that knowing the decay of a function $f$ on a discrete set $Λ\subset\mathbb{R}$ and the decay of its Fourier transform $\hat{f}$ on a discrete set $M\subset\mathbb{R}$ is enough to determine the global decay of $f$ and $\hat{f}$, provided that $(Λ,M)$ is a supercritical pair in the sense of Kulikov, Nazarov, and Sodin. This decay transfer result leads to a discrete generalization of Morgan's uncertainty principle: it is enough to require $|f(λ)|\lesssim e^{-\frac{2}{p}Aπ|λ|^p}$ for all $λ\inΛ$ and $|\hat{f}(μ)|\lesssim e^{-\frac{2}{q}Aπ|μ|^q}$ for all $μ\in M$, where $(p,q)$ are Hölder conjugates, $A>|\cos(\frac{rπ}{2})|^\frac{1}{r}$, and $r:=\min\{p,q\}$. For $A=1$ and $p,q=2$, we also show that any such function must be a scaled Gaussian. This yields a discrete version of Hardy's uncertainty principle and resolves two questions posed by Ramos and Sousa. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_03679 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A discrete Hardy uncertainty principle Lysen, Torgeir Keun Classical Analysis and ODEs Complex Variables 42A38 We show that knowing the decay of a function $f$ on a discrete set $Λ\subset\mathbb{R}$ and the decay of its Fourier transform $\hat{f}$ on a discrete set $M\subset\mathbb{R}$ is enough to determine the global decay of $f$ and $\hat{f}$, provided that $(Λ,M)$ is a supercritical pair in the sense of Kulikov, Nazarov, and Sodin. This decay transfer result leads to a discrete generalization of Morgan's uncertainty principle: it is enough to require $|f(λ)|\lesssim e^{-\frac{2}{p}Aπ|λ|^p}$ for all $λ\inΛ$ and $|\hat{f}(μ)|\lesssim e^{-\frac{2}{q}Aπ|μ|^q}$ for all $μ\in M$, where $(p,q)$ are Hölder conjugates, $A>|\cos(\frac{rπ}{2})|^\frac{1}{r}$, and $r:=\min\{p,q\}$. For $A=1$ and $p,q=2$, we also show that any such function must be a scaled Gaussian. This yields a discrete version of Hardy's uncertainty principle and resolves two questions posed by Ramos and Sousa. |
| title | A discrete Hardy uncertainty principle |
| topic | Classical Analysis and ODEs Complex Variables 42A38 |
| url | https://arxiv.org/abs/2605.03679 |