A complex-variable proof of the Wiener tauberian theorem
Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 91-96.

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.

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.

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     title = {A complex-variable proof of the {Wiener} tauberian theorem},
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Esterlé, Jean. A complex-variable proof of the Wiener tauberian theorem. Annales de l'Institut Fourier, Volume 30 (1980) no. 2, pp. 91-96. doi : 10.5802/aif.786. http://www.numdam.org/articles/10.5802/aif.786/

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