Some properties of a modified Hilbert transform

Matteo Ferrari

doi:10.5802/crmath.600

Research article - Numerical analysis

Some properties of a modified Hilbert transform

Matteo Ferrari ¹

¹ University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 799-806.

Abstract
Résumé

Recently, Steinbach et al. introduced a novel operator $ℋ_{T} : L^{2} (0, T) \to L^{2} (0, T)$ , 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 $ℋ_{T}$ and the canonical Hilbert transform $ℋ$ . Specifically, we prove the relationship $ℋ_{T} φ = - ℋ \tilde{φ}$ , where $φ \in L^{2} (0, T)$ and $\tilde{φ}$ is a suitable extension of $φ$ over the entire $ℝ$ . By leveraging this crucial result, we derive some properties of $ℋ_{T}$ , including a new inversion formula, that emerge as immediate consequences of well-established findings on $ℋ$ .

Récemment, Steinbach et al. ont introduit un nouvel opérateur $ℋ_{T} : L^{2} (0, T) \to L^{2} (0, T)$ , connu sous le nom de transformée de Hilbert modifiée. Cet opérateur a montré son importance dans les formulations spatio-temporelles liées aux équations de la chaleur et des ondes. Dans cet article, nous établissons un lien direct entre la transformée de Hilbert modifiée $ℋ_{T}$ et la transformée de Hilbert canonique $ℋ$ . Plus précisément, nous prouvons la relation $ℋ_{T} φ = - ℋ \tilde{φ}$ , où $φ \in L^{2} (0, T)$ et $\tilde{φ}$ est une extension appropriée de $φ$ sur l’ensemble de $ℝ$ . En tirant parti de ce résultat crucial, nous déduisons certaines propriétés de $ℋ_{T}$ , y compris une nouvelle formule d’inversion, qui émergent comme des conséquences immédiates de résultats bien établis sur $ℋ$ .

Received: 2023-09-25
Accepted: 2023-12-15
Published online: 2024-09-17

DOI: 10.5802/crmath.600

Classification: 44A15

Author's affiliations:

Matteo Ferrari ¹

¹ University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Matteo Ferrari},
     title = {Some properties of a modified {Hilbert} transform},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {799--806},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.600},
     language = {en},
}

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Matteo Ferrari. Some properties of a modified Hilbert transform. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 799-806. doi : 10.5802/crmath.600. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.600/

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