no. 2
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.

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, {x∈R| 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 {x∈R| 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.

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     title = {A property of {Fourier} {Stieltjes} transforms on the discrete group of real numbers},
     journal = {Annales de l'Institut Fourier},
     pages = {325--334},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {20},
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     year = {1970},
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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/articles/10.5802/aif.356/

[1] H. Rosenthal, A characterization of restrictions of Fourier-Stieltjes transforms, Pac. J. Math. 23 (1967) 403-418. | MR | Zbl

[2] W. Rudin, Fourier analysis on groups. New York 1962. | MR | Zbl

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