A property of Fourier Stieltjes transforms on the discrete group of real numbers
Annales de l'Institut Fourier, Volume 20 (1970) no. 2, p. 325-334
Let μ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on R. Then for every ε>0, {xR| Re (μ(x))>ε} has a vanishing interior Lebesgue measure. If ε=0 the statement is not generally true. The result is applied to prove a theorem of Rosenthal.
Soit μ une transformée de Fourier-Stieltjes, définie sur la droite réelle discrète et avec la mesure correspondante sur le groupe dual s’annulant sur l’ensemble des caractères continus sur R. Alors pour chaque ε>0 la mesure de Lebesgue intérieure de {xR| Re (μ(x))>ε} est nulle. Pour ε=0 la proposition est, en général, inexacte. Le résultat est appliqué pour démontrer un théorème de M. Rosenthal.
@article{AIF_1970__20_2_325_0,
     author = {Domar, Yngve},
     title = {A property of Fourier Stieltjes transforms on the discrete group of real numbers},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {20},
     number = {2},
     year = {1970},
     pages = {325-334},
     doi = {10.5802/aif.356},
     zbl = {0183.40002},
     mrnumber = {44 \#3077},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1970__20_2_325_0}
}
Domar, Yngve. A property of Fourier Stieltjes transforms on the discrete group of real numbers. Annales de l'Institut Fourier, Volume 20 (1970) no. 2, pp. 325-334. doi : 10.5802/aif.356. http://www.numdam.org/item/AIF_1970__20_2_325_0/

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