no. 17-18
Harmonic Analysis/Functional Analysis
A simple real-variable proof that the Hilbert transform is an L2-isometry
[Une démonstration simple en variables réelles de la propriété d'isométrie L2 de la transformation de Hilbert H]
Présenté par : Yves Meyer
Laeng, Enrico1
1 Politecnico di Milano, Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy
Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 977-980

The Hilbert transform H can be extended to an isometry of L2. We prove this fact working directly on the principal value integral, completely avoiding the use of the Fourier transform and the use of orthogonal systems of functions. Our approach here is a byproduct of our attempts to understand the rearrangement properties of H.

La transformation de Hilbert H peut être étendue à une isometrie dans L2. On demontre cette propriété en utilsant directement la valeur principale de l'intégrale, sans utiliser la transformation de Fourier, ni des systèmes de fonctions orthogonales. L'approche proposée est liée à nos tentative de comprendre le proprietés de réarrangement de H.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.002

Laeng, Enrico 1

1 Politecnico di Milano, Dipartimento di Matematica “F. Brioschi”, Via Bonardi 9, 20133 Milano, Italy 
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Laeng, Enrico. A simple real-variable proof that the Hilbert transform is an $ {L}^{2}$-isometry. Comptes Rendus. Mathématique, Tome 348 (2010) no. 17-18, pp. 977-980. doi: 10.1016/j.crma.2010.07.002

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