A complex-variable proof of the Wiener tauberian theorem
Annales de l'Institut Fourier, Tome 30 (1980) no. 2, p. 91-96
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},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {30},
     number = {2},
     year = {1980},
     pages = {91-96},
     doi = {10.5802/aif.786},
     zbl = {0419.40005},
     mrnumber = {81j:43016},
     language = {en},
     url = {http://http://www.numdam.org/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) no. 2, pp. 91-96. doi : 10.5802/aif.786. http://www.numdam.org/item/AIF_1980__30_2_91_0/

[1] R. P. Boas, Entire functions, Academic press, New-York, 1954. | MR 16,914f | Zbl 0058.30201

[2] P. Eymard, L'algèbre de Fourier d'un groupe localement compact, Bull. Soc. Math. de France, 92 (1964), 181-236. | Numdam | MR 37 #4208 | Zbl 0169.46403

[3] H. Leptin, On group algebras of nilpotent groups, Studia Math., 47 (1973), 37-49. | MR 48 #9262 | Zbl 0258.22009

[4] H. Leptin, Ideal theory in group algebras of locally compact groups, Inventiones Math., 31 (1976), 259-278. | MR 53 #3189 | Zbl 0328.22012

[5] A. M. Sinclair, Factorization, bounded approximate identities and a convolution algebra, J. Func. An. | Zbl 0385.46030