no. 4
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},
     year = {1973},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {23},
     number = {4},
     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

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[2] J. Benedetto, Harmonic Analysis on Totally Disconnected Sets, Lecture Notes in Mathematics 202, Springer-Verlag, New York (1971). | Zbl | MR

[3] J. Benedetto, Dirichlet Series, Spectral Synthesis, and Algebraic Number Fields, U. of Maryland Mathematics Dept. TR 71-41 (1971), 1-23.

[4] J. W. S. Cassels and A. Fröhlich, (editors) Algebraic Number Theory, Thompson Book Company, Washington, D. C. (1967).

[5] L. J. Goldstein, Analytic Number Theory Prentice, Hall, N. J. (1971). | Zbl | MR

[6] J.-P. Kahane, Séries de Fourier absolument convergentes, Springer-Verlag, New York (1970). | Zbl | MR

[7] I. Richards, «On the Disproof of Spectral Synthesis» J. of Comb. Theory 2 (1967), 61-70. | Zbl | MR

[8] A. Weil, Basic Number Theory, Springer-Verlag, New York (1967). | Zbl

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