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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| Format: | Article |
| Language: | English |
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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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| Summary: | 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 $\mathcal{H}$. |
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| ISSN: | 1778-3569 |