# A property of Fourier Stieltjes transforms on the discrete group of real numbers

• Volume: 20, Issue: 2, page 325-334
• ISSN: 0373-0956

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## Abstract

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Let $\mu$ 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 $\mathbf{R}$. Then for every $ϵ>0$, $\left\{x\in \mathbf{R}|\phantom{\rule{0.166667em}{0ex}}\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\left(\mu \left(x\right)\right)>ϵ\right\}$ 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.

## How to cite

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Domar, Yngve. "A property of Fourier Stieltjes transforms on the discrete group of real numbers." Annales de l'institut Fourier 20.2 (1970): 325-334. <http://eudml.org/doc/74018>.

@article{Domar1970,
abstract = {Let $\mu$ 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 $\{\bf R\}$. Then for every $\varepsilon &gt;0$, $\lbrace x\in \{\bf R\}\vert \, \{\rm Re\}\, (\mu (x))&gt;\varepsilon \rbrace$ has a vanishing interior Lebesgue measure. If $\varepsilon =0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.},
author = {Domar, Yngve},
journal = {Annales de l'institut Fourier},
keywords = {integral equations, integral transforms},
language = {eng},
number = {2},
pages = {325-334},
publisher = {Association des Annales de l'Institut Fourier},
title = {A property of Fourier Stieltjes transforms on the discrete group of real numbers},
url = {http://eudml.org/doc/74018},
volume = {20},
year = {1970},
}

TY - JOUR
AU - Domar, Yngve
TI - A property of Fourier Stieltjes transforms on the discrete group of real numbers
JO - Annales de l'institut Fourier
PY - 1970
PB - Association des Annales de l'Institut Fourier
VL - 20
IS - 2
SP - 325
EP - 334
AB - Let $\mu$ 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 ${\bf R}$. Then for every $\varepsilon &gt;0$, $\lbrace x\in {\bf R}\vert \, {\rm Re}\, (\mu (x))&gt;\varepsilon \rbrace$ has a vanishing interior Lebesgue measure. If $\varepsilon =0$ the statement is not generally true. The result is applied to prove a theorem of Rosenthal.
LA - eng
KW - integral equations, integral transforms
UR - http://eudml.org/doc/74018
ER -

## References

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1. [1] H. ROSENTHAL, A characterization of restrictions of Fourier-Stieltjes transforms, Pac. J. Math. 23 (1967) 403-418. Zbl0155.18901MR36 #3065
2. [2] W. RUDIN, Fourier analysis on groups. New York 1962. Zbl0107.09603MR27 #2808

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