Idele characters in spectral synthesis on 𝐑/2Ï€đ™
Annales de l'Institut Fourier, Volume 23 (1973) no. 4, p. 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 sĂ©rie de Dirichlet F(x,x) oĂč pour chaque s fixĂ© dans un demi-plan, Re F(x,x), comme une fonction de x, est une sĂ©rie de Fourier absolument convergente qui ne satisfait pas Ă  la synthĂšse. À cause de la mĂ©thode de choix des frĂ©quences en F, nous sommes conduits Ă  introduire une classe de sĂ©ries de Fourier absolument convergentes, satisfaisant Ă  la synthĂšse, qui sont dĂ©finis par les caractĂšres idĂšles. Nous rĂ©soudrons “le problĂšme du prolongement analytique” dans ce milieu en construisant les pseudomesures dĂ©terminĂ©es par les caractĂšres idĂšles, quand Re s≀1.

     author = {Benedetto, John J.},
     title = {Idele characters in spectral synthesis on ${\bf R}/2\pi {\bf Z}$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {23},
     number = {4},
     year = {1973},
     pages = {45-64},
     doi = {10.5802/aif.481},
     zbl = {0253.12016},
     mrnumber = {50 \#14068},
     language = {en},
     url = {}
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.

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