Siegel-Radon transforms of transverse dynamical systems
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918014933270528 |
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| author | Björklund, Michael Hartnick, Tobias |
| author_facet | Björklund, Michael Hartnick, Tobias |
| contents | We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $Θ$-) transforms and Marklof-Strömbergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis.
Our main applications concern dynamical systems $(X, μ)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, μ)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case.
In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schrödinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_05980 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Siegel-Radon transforms of transverse dynamical systems Björklund, Michael Hartnick, Tobias Dynamical Systems Group Theory Number Theory We extend Helgason's classical definition of a generalized Radon transform, defined for a pair of homogeneous spaces of an lcsc group $G$, to a broader setting in which one of the spaces is replaced by a possibly non-homogeneous dynamical system over $G$ together with a suitable cross section. This general framework encompasses many examples studied in the literature, including Siegel (or $Θ$-) transforms and Marklof-Strömbergsson transforms in the geometry of numbers, Siegel-sVeech transforms for translation surfaces, and Zak transforms in time-frequency analysis. Our main applications concern dynamical systems $(X, μ)$ in which the cross section is induced from a separated cross section. We establish criteria for the boundedness, integrability, and square-integrability of the associated Siegel-Radon transforms, and show how these transforms can be used to embed induced $G$-representations into $L^p(X, μ)$ for appropriate values of $p$. These results apply in particular to hulls of approximate lattices and certain "thinnings" thereof, including arbitrary positive density subsets in the amenable case. In the special case of cut-and-project sets, we derive explicit formulas for the dual transforms, and in the special case of the Heisenberg group we provide isometric embedding of Schrödinger representations into the $L^2$-space of the hulls of positive density subsets of approximate lattices in the Heisenberg group by means of aperiodic Zak transforms. |
| title | Siegel-Radon transforms of transverse dynamical systems |
| topic | Dynamical Systems Group Theory Number Theory |
| url | https://arxiv.org/abs/2505.05980 |