Some properties of a modified Hilbert transform
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914739658948608 |
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| author | Ferrari, Matteo |
| author_facet | Ferrari, Matteo |
| contents | Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, 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 $\mathcal{H}_T φ= -\mathcal{H} \tildeφ$, where $φ\in L^2(0,T)$ and $\tildeφ$ is a suitable extension of $φ$ 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 | Preprint |
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arxiv_https___arxiv_org_abs_2404_02609 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some properties of a modified Hilbert transform Ferrari, Matteo Classical Analysis and ODEs Numerical Analysis 44A15 Recently, Steinbach et al. introduced a novel operator $\mathcal{H}_T: L^2(0,T) \to L^2(0,T)$, 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 $\mathcal{H}_T φ= -\mathcal{H} \tildeφ$, where $φ\in L^2(0,T)$ and $\tildeφ$ is a suitable extension of $φ$ 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 $\mathcal{H}$. |
| title | Some properties of a modified Hilbert transform |
| topic | Classical Analysis and ODEs Numerical Analysis 44A15 |
| url | https://arxiv.org/abs/2404.02609 |