A simple real-variable proof that the Hilbert transform is an L2 -isometry
Comptes Rendus Mathematique, ISSN: 1631-073X, Vol: 348, Issue: 17, Page: 977-980
2010
- 2Citations
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Article Description
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.
Bibliographic Details
http://www.sciencedirect.com/science/article/pii/S1631073X10001913; http://dx.doi.org/10.1016/j.crma.2010.07.002; http://www.scopus.com/inward/record.url?partnerID=HzOxMe3b&scp=77957141652&origin=inward; https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.002/; https://dx.doi.org/10.1016/j.crma.2010.07.002
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