# Idele characters in spectral synthesis on $\mathbf{R}/2\pi \mathbf{Z}$

Annales de l'institut Fourier (1973)

- Volume: 23, Issue: 4, page 45-64
- ISSN: 0373-0956

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topBenedetto, John J.. "Idele characters in spectral synthesis on ${\bf R}/2\pi {\bf Z}$." Annales de l'institut Fourier 23.4 (1973): 45-64. <http://eudml.org/doc/74153>.

@article{Benedetto1973,

abstract = {Let $s\in \{\bf C\}$, $x\in \{\bf R\}/2\pi \{\bf Z\}$. We construct Dirichlet series $F(x,x)$ where for each fixed $s$ in a half plane, $\{\rm Re\}\, F(x,x)$, 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 $\{\rm Re\}\, s\le 1$.},

author = {Benedetto, John J.},

journal = {Annales de l'institut Fourier},

language = {eng},

number = {4},

pages = {45-64},

publisher = {Association des Annales de l'Institut Fourier},

title = {Idele characters in spectral synthesis on $\{\bf R\}/2\pi \{\bf Z\}$},

url = {http://eudml.org/doc/74153},

volume = {23},

year = {1973},

}

TY - JOUR

AU - Benedetto, John J.

TI - Idele characters in spectral synthesis on ${\bf R}/2\pi {\bf Z}$

JO - Annales de l'institut Fourier

PY - 1973

PB - Association des Annales de l'Institut Fourier

VL - 23

IS - 4

SP - 45

EP - 64

AB - Let $s\in {\bf C}$, $x\in {\bf R}/2\pi {\bf Z}$. We construct Dirichlet series $F(x,x)$ where for each fixed $s$ in a half plane, ${\rm Re}\, F(x,x)$, 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 ${\rm Re}\, s\le 1$.

LA - eng

UR - http://eudml.org/doc/74153

ER -

## References

top- [1] J. BENEDETTO, Harmonic Synthesis and Pseudo-Measures, U. of Maryland Mathematics Dept. Lecture Notes No. 5 (1968).
- [2] J. BENEDETTO, Harmonic Analysis on Totally Disconnected Sets, Lecture Notes in Mathematics 202, Springer-Verlag, New York (1971). Zbl0225.43001MR56 #6287
- [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). Zbl0226.12001MR58 #16471
- [6] J.-P. KAHANE, Séries de Fourier absolument convergentes, Springer-Verlag, New York (1970). Zbl0195.07602MR43 #801
- [7] I. RICHARDS, «On the Disproof of Spectral Synthesis» J. of Comb. Theory 2 (1967), 61-70. Zbl0147.33802MR34 #4807
- [8] A. WEIL, Basic Number Theory, Springer-Verlag, New York (1967). Zbl0176.33601

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