La transformation de Hilbert H peut être étendue à une isometrie dans . 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.
The Hilbert transform H can be extended to an isometry of . 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.
Accepté le :
Publié le :
@article{CRMATH_2010__348_17-18_977_0, author = {Laeng, Enrico}, title = {A simple real-variable proof that the {Hilbert} transform is an $ {L}^{2}$-isometry}, journal = {Comptes Rendus. Math\'ematique}, pages = {977--980}, publisher = {Elsevier}, volume = {348}, number = {17-18}, year = {2010}, doi = {10.1016/j.crma.2010.07.002}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2010.07.002/} }
TY - JOUR AU - Laeng, Enrico TI - A simple real-variable proof that the Hilbert transform is an $ {L}^{2}$-isometry JO - Comptes Rendus. Mathématique PY - 2010 SP - 977 EP - 980 VL - 348 IS - 17-18 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2010.07.002/ DO - 10.1016/j.crma.2010.07.002 LA - en ID - CRMATH_2010__348_17-18_977_0 ER -
%0 Journal Article %A Laeng, Enrico %T A simple real-variable proof that the Hilbert transform is an $ {L}^{2}$-isometry %J Comptes Rendus. Mathématique %D 2010 %P 977-980 %V 348 %N 17-18 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2010.07.002/ %R 10.1016/j.crma.2010.07.002 %G en %F CRMATH_2010__348_17-18_977_0
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. http://archive.numdam.org/articles/10.1016/j.crma.2010.07.002/
[1] Some sharp inequalities for conjugate functions, Indiana University Mathematics Journal, Volume 27 (1978) no. 5, pp. 833-852
[2] Variations on a theme of Boole and Stein–Weiss, Journal of Mathematical Analysis and Applications, Volume 363 (2010), pp. 225-229
[3] The Hilbert transform and Hermite functions: A real variable proof of the -isometry, Journal of Mathematical Analysis and Applications, Volume 347 (2008), pp. 592-596
[4] Sharp -estimates for the segment multiplier, Collectanea Mathematica, Volume 51 (2000) no. 3, pp. 309-326
Cité par Sources :