no. 2
A complex-variable proof of the Wiener tauberian theorem
Annales de l'Institut Fourier, Tome 30 (1980) no. 2, pp. 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 Ret≥1. 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 Ret≥1. 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},
     pages = {91--96},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {30},
     number = {2},
     year = {1980},
     doi = {10.5802/aif.786},
     zbl = {0419.40005},
     mrnumber = {81j:43016},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.786/}
}
TY  - JOUR
AU  - Esterlé, Jean
TI  - A complex-variable proof of the Wiener tauberian theorem
JO  - Annales de l'Institut Fourier
PY  - 1980
SP  - 91
EP  - 96
VL  - 30
IS  - 2
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.786/
DO  - 10.5802/aif.786
LA  - en
ID  - AIF_1980__30_2_91_0
ER  - 
%0 Journal Article
%A Esterlé, Jean
%T A complex-variable proof of the Wiener tauberian theorem
%J Annales de l'Institut Fourier
%D 1980
%P 91-96
%V 30
%N 2
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.786/
%R 10.5802/aif.786
%G en
%F 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. https://aif.centre-mersenne.org/articles/10.5802/aif.786/

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

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

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

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

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

Cité par Sources :