Esterlè's proof of the tauberian theorem for Beurling algebras
Annales de l'Institut Fourier, Volume 31 (1981) no. 4, p. 141-150

Recently in this Journal J. Esterlé gave a new proof of the Wiener Tauberian theorem for L 1 (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 ϕ 1 (R). Our estimates need a theorem of Hayman and Korenblum.

Récemment dans ce Journal J. Esterlé a donné une preuve nouvelle du théorème taubérien de Wiener pour L 1 (R) en utilisant le théorème de Ahlfors-Heins pour les fonctions analytiques bornées sur un demi-plan. Ici nous utilisons essentiellement la même méthode pour certaines algèbres de Beurling L ϕ 1 (R). Nos évaluations ont besoin d’un théorème de Hayman et Korenblum.

@article{AIF_1981__31_4_141_0,
     author = {Dales, H. G. and Hayman, W. K.},
     title = {Esterl\`e's proof of the tauberian theorem for Beurling algebras},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {31},
     number = {4},
     year = {1981},
     pages = {141-150},
     doi = {10.5802/aif.852},
     zbl = {0449.40005},
     mrnumber = {83j:43007},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1981__31_4_141_0}
}
Dales, H. G.; Hayman, W. K. Esterlè's proof of the tauberian theorem for Beurling algebras. Annales de l'Institut Fourier, Volume 31 (1981) no. 4, pp. 141-150. doi : 10.5802/aif.852. http://www.numdam.org/item/AIF_1981__31_4_141_0/

[1] A. Beurling, Sur les intégrales de Fourier absolument convergentes et leur application à une transformation fonctionnelle, Neuvième Congr. Math. Scandinaves, (Helsinki, 1938), Tryekeri, Helsinki (1939), 199-210. | JFM 65.0483.02

[2] A. Beurling and P. Malliavin, The Fourier transforms of measures with compact support, Acta Math., 107 (1962), 291-309. | MR 26 #5361 | Zbl 0127.32601

[3] R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954. | MR 16,914f | Zbl 0058.30201

[4] Y. Domar, Translation invariant subspaces of weighted lp and Lp spaces, Math. Scand., 49 (1981), to appear. | MR 83k:47022 | Zbl 0465.47020

[5] J. Esterle, A complex-variable proof of the Wiener Tauberian theorem, Ann. Inst. Fourier, Grenoble, 30 (1980), 91-96. | Numdam | MR 81j:43016 | Zbl 0419.40005

[6] I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative normed rings, Chelsea Publishing Co., New York, 1964.

[7] V. P. Gurarii, Harmonic analysis in spaces with a weight, Trudy Moskov. Mat. Obšč., 36 (1976), 21-76. = Trans. Moscow Math. Soc., 35 (1979), 21-75. | MR 58 #17684 | Zbl 0425.43007

[8] W. K. Hayman and B. Korenblum, An extension of the Riesz-Herglotz formula, Annales Academiae Scientiarum Fennicae, Series A1, Mathematica, 2 (1976), 175-201. | MR 57 #6446 | Zbl 0416.30019

[9] B. Korenblum, A generalization of Wiener's Tauberian theorem and harmonic analysis of rapidly increasing functions, (Russian), Trudy Moskow. Mat. Obšč., 7 (1958), 121-148.

[10] R.E.A.C. Paley and N. Wiener, Fourier transforms in the complex domain, American Math. Soc. Colloquium Publications, XIX, New York, 1934. | JFM 60.0345.02 | Zbl 0011.01601

[11] A. Vretblad, Spectral analysis in weighted L1 spaces on R, Ark. Math., 11 (1973), 109-138. | MR 50 #5361 | Zbl 0258.46047