Idele characters in spectral synthesis on 𝐑/2π𝐙
Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 45-64.

Let s∈C, x∈R/2πZ. We construct Dirichlet series F(x,x) where for each fixed s in a half plane, Re F(x,x), as a function of x, is a non-synthesizable absolutely convergent Fourier series. Because of the way the frequencies in F are chosen, we are motivated to introduce a class of synthesizable absolutely convergent Fourier series which are defined in terms of idele characters. We solve the “problem of analytic continuation” in this setting by constructing pseudo-measures, determined by idele characters, when Re s≤1.

Soit s∈C, x∈R/2πZ. Nous construisons une série de Dirichlet F(x,x) où pour chaque s fixé dans un demi-plan, Re F(x,x), comme une fonction de x, est une série de Fourier absolument convergente qui ne satisfait pas à la synthèse. À cause de la méthode de choix des fréquences en F, nous sommes conduits à introduire une classe de séries de Fourier absolument convergentes, satisfaisant à la synthèse, qui sont définis par les caractères idèles. Nous résoudrons “le problème du prolongement analytique” dans ce milieu en construisant les pseudomesures déterminées par les caractères idèles, quand Re s≤1.

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     author = {Benedetto, John J.},
     title = {Idele characters in spectral synthesis on ${\bf R}/2\pi {\bf Z}$},
     journal = {Annales de l'Institut Fourier},
     pages = {45--64},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {23},
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     year = {1973},
     doi = {10.5802/aif.481},
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Benedetto, John J. Idele characters in spectral synthesis on ${\bf R}/2\pi {\bf Z}$. Annales de l'Institut Fourier, Volume 23 (1973) no. 4, pp. 45-64. doi : 10.5802/aif.481. https://aif.centre-mersenne.org/articles/10.5802/aif.481/

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