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| Format: | Preprint |
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2024
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| Accès en ligne: | https://arxiv.org/abs/2410.04557 |
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| _version_ | 1866929529981763584 |
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| author | Radchenko, Danylo Ramos, João P. G. |
| author_facet | Radchenko, Danylo Ramos, João P. G. |
| contents | This work focuses on two questions raised by H. Hedenmalm and A. Montes-Rodríguez on Heisenberg Uniqueness Pairs for perturbed lattice crosses.
The first of them deals with a complete characterization of $β>0$ for which, for a fixed $θ\in \mathbb{R},$ the translated lattice cross $Λ_β^θ = ((\mathbb{Z} + \{θ\}) \times \{0\}) \cup (\{0\} \times β\mathbb{Z})$ satisfies that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair, where $Γ$ is the hyperbola in $\mathbb{R}^2$ with axes as asymptotes. We show that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair if and only if $β\le 1$, confirming a prediction made by Hedenmalm and Montes-Rodríguez.
Furthermore, under modified decay conditions on the measures under consideration, we are able to prove sharp results for when a perturbed lattice cross $Λ_{\bf A,B}$ is such that $(Γ,Λ_{\bf A,B})$ is a Heisenberg Uniqueness Pair. In particular, under such decay conditions, this solves another question posed by Hedenmalm and Montes-Rodríguez.
Our techniques run through the analysis of the action of the operator that maps the Fourier transform of an $L^1$ function $ψ$ to the Fourier transform of $t^{-2} ψ(1/t)$. In other words, we analyze the operator taking the restriction to the $x$-axis of a solution $u$ to the Klein-Gordon equation to its restriction to the $y$-axis. This operator turns out to be related to the action of the four-dimensional Fourier transform on radial functions, which enables us to use the framework and techniques of discrete uncertainty principles for the Fourier transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04557 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perturbed lattice crosses and Heisenberg uniqueness pairs Radchenko, Danylo Ramos, João P. G. Classical Analysis and ODEs Functional Analysis 35A02, 42B10 This work focuses on two questions raised by H. Hedenmalm and A. Montes-Rodríguez on Heisenberg Uniqueness Pairs for perturbed lattice crosses. The first of them deals with a complete characterization of $β>0$ for which, for a fixed $θ\in \mathbb{R},$ the translated lattice cross $Λ_β^θ = ((\mathbb{Z} + \{θ\}) \times \{0\}) \cup (\{0\} \times β\mathbb{Z})$ satisfies that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair, where $Γ$ is the hyperbola in $\mathbb{R}^2$ with axes as asymptotes. We show that $(Γ,Λ_β^θ)$ is a Heisenberg Uniqueness Pair if and only if $β\le 1$, confirming a prediction made by Hedenmalm and Montes-Rodríguez. Furthermore, under modified decay conditions on the measures under consideration, we are able to prove sharp results for when a perturbed lattice cross $Λ_{\bf A,B}$ is such that $(Γ,Λ_{\bf A,B})$ is a Heisenberg Uniqueness Pair. In particular, under such decay conditions, this solves another question posed by Hedenmalm and Montes-Rodríguez. Our techniques run through the analysis of the action of the operator that maps the Fourier transform of an $L^1$ function $ψ$ to the Fourier transform of $t^{-2} ψ(1/t)$. In other words, we analyze the operator taking the restriction to the $x$-axis of a solution $u$ to the Klein-Gordon equation to its restriction to the $y$-axis. This operator turns out to be related to the action of the four-dimensional Fourier transform on radial functions, which enables us to use the framework and techniques of discrete uncertainty principles for the Fourier transform. |
| title | Perturbed lattice crosses and Heisenberg uniqueness pairs |
| topic | Classical Analysis and ODEs Functional Analysis 35A02, 42B10 |
| url | https://arxiv.org/abs/2410.04557 |