Esterlè's proof of the tauberian theorem for Beurling algebras

Dales, H.G. and Hayman, W. K. (1981) Esterlè's proof of the tauberian theorem for Beurling algebras. Annales de L'Institut Fourier, 31 (4). pp. 141-150. ISSN 1777-5310

Abstract

Recently in this Journal J. Esterlé gave a new proof of the Wiener Tauberian theorem for $L^1({\bf R})$ using the Ahlfors-Heins theorem for bounded analytic functions on a half-plane. We here use essentially the same method to prove the analogous result for Beurling algebras $L^1_\varphi ({\bf R})$. Our estimates need a theorem of Hayman and Korenblum.