A complex-variable proof of the Wiener tauberian theorem
Esterlé, Jean
Annales de l'Institut Fourier, Tome 30 (1980), p. 91-96 / Harvested from Numdam

Le semi-groupe fondamental (a t ) t>0 de l’équation de la chaleur pour la droite réelle possède une extension analytique (a t ) Re t>0 au demi-plan droit qui vérifie a t |t| pour Ret1. 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 (a t ) t>0 of the heat equation for the real line has an analytic extension (a t ) Re t>0 to the right-hand open half plane which satisfies a t |t| for Ret1. Using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane we show that the Wiener tauberian theorem for L 1 (R) 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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