A complex-variable proof of the Wiener tauberian theorem

Esterlé, Jean. "A complex-variable proof of the Wiener tauberian theorem." Annales de l'institut Fourier 30.2 (1980): 91-96. <http://eudml.org/doc/74452>.

@article{Esterlé1980,
abstract = {The fundamental semigroup $(a^t)_\{t&gt;0\}$ of the heat equation for the real line has an analytic extension $(a^t)_\{\{\rm Re\}\, t&gt;0\}$ to the right-hand open half plane which satisfies $\Vert a^t\Vert \le \sqrt\{\vert t\vert \}$ for Re$\, t\ge 1$. Using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane we show that the Wiener tauberian theorem for $L^1(\{\bf R\})$ follows from the above inequality.},
author = {Esterlé, Jean},
journal = {Annales de l'institut Fourier},
keywords = {Wiener tauberian theorem},
language = {eng},
number = {2},
pages = {91-96},
publisher = {Association des Annales de l'Institut Fourier},
title = {A complex-variable proof of the Wiener tauberian theorem},
url = {http://eudml.org/doc/74452},
volume = {30},
year = {1980},
}

TY - JOUR
AU - Esterlé, Jean
TI - A complex-variable proof of the Wiener tauberian theorem
JO - Annales de l'institut Fourier
PY - 1980
PB - Association des Annales de l'Institut Fourier
VL - 30
IS - 2
SP - 91
EP - 96
AB - The fundamental semigroup $(a^t)_{t&gt;0}$ of the heat equation for the real line has an analytic extension $(a^t)_{{\rm Re}\, t&gt;0}$ to the right-hand open half plane which satisfies $\Vert a^t\Vert \le \sqrt{\vert t\vert }$ for Re$\, t\ge 1$. Using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane we show that the Wiener tauberian theorem for $L^1({\bf R})$ follows from the above inequality.
LA - eng
KW - Wiener tauberian theorem
UR - http://eudml.org/doc/74452
ER -