Some properties of a modified Hilbert transform
Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \rightarrow L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection...
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Académie des sciences
2024-09-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.600/ |
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| author | Ferrari, Matteo |
| author_facet | Ferrari, Matteo |
| author_sort | Ferrari, Matteo |
| collection | DOAJ |
| description | Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \rightarrow L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert transform $\mathcal{H}_T$ and the canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the relationship $\mathcal{H}_T \varphi = -\mathcal{H} \widetilde{\varphi }$, where $\varphi \in L^2(0,T)$ and $\widetilde{\varphi }$ is a suitable extension of $\varphi $ over the entire $\mathbb{R}$. By leveraging this crucial result, we derive some properties of $\mathcal{H}_T$, including a new inversion formula, that emerge as immediate consequences of well-established findings on $\mathcal{H}$. |
| format | Article |
| id | doaj-art-39355d9bd686468ba4dfcc8683a89a2e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2024-09-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-39355d9bd686468ba4dfcc8683a89a2e2025-02-07T11:22:28ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-09-01362G779980610.5802/crmath.60010.5802/crmath.600Some properties of a modified Hilbert transformFerrari, Matteo0https://orcid.org/0000-0002-2577-1421University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090, Vienna, AustriaRecently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \rightarrow L^2(0,T)$, known as the modified Hilbert transform. This operator has shown its significance in space-time formulations related to the heat and wave equations. In this paper, we establish a direct connection between the modified Hilbert transform $\mathcal{H}_T$ and the canonical Hilbert transform $\mathcal{H}$. Specifically, we prove the relationship $\mathcal{H}_T \varphi = -\mathcal{H} \widetilde{\varphi }$, where $\varphi \in L^2(0,T)$ and $\widetilde{\varphi }$ is a suitable extension of $\varphi $ over the entire $\mathbb{R}$. By leveraging this crucial result, we derive some properties of $\mathcal{H}_T$, including a new inversion formula, that emerge as immediate consequences of well-established findings on $\mathcal{H}$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.600/ |
| spellingShingle | Ferrari, Matteo Some properties of a modified Hilbert transform Comptes Rendus. Mathématique |
| title | Some properties of a modified Hilbert transform |
| title_full | Some properties of a modified Hilbert transform |
| title_fullStr | Some properties of a modified Hilbert transform |
| title_full_unstemmed | Some properties of a modified Hilbert transform |
| title_short | Some properties of a modified Hilbert transform |
| title_sort | some properties of a modified hilbert transform |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.600/ |
| work_keys_str_mv | AT ferrarimatteo somepropertiesofamodifiedhilberttransform |