Le semi-groupe fondamental de l’équation de la chaleur pour la droite réelle possède une extension analytique au demi-plan droit qui vérifie pour Re. En utilisant le théorème de Ahlfors-Heins pour les fonctions analytiques bornées sur le demi-plan on peut déduire le théorème taubérien de Wiener de l’inégalité ci-dessus.
The fundamental semigroup of the heat equation for the real line has an analytic extension to the right-hand open half plane which satisfies for Re. Using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane we show that the Wiener tauberian theorem for follows from the above inequality.
@article{AIF_1980__30_2_91_0, author = {Esterl\'e, Jean}, title = {A complex-variable proof of the Wiener tauberian theorem}, journal = {Annales de l'Institut Fourier}, volume = {30}, year = {1980}, pages = {91-96}, doi = {10.5802/aif.786}, mrnumber = {81j:43016}, zbl = {0419.40005}, language = {en}, url = {http://dml.mathdoc.fr/item/AIF_1980__30_2_91_0} }
Esterlé, Jean. A complex-variable proof of the Wiener tauberian theorem. Annales de l'Institut Fourier, Tome 30 (1980) pp. 91-96. doi : 10.5802/aif.786. http://gdmltest.u-ga.fr/item/AIF_1980__30_2_91_0/
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