[Régularité maximale de la partie anti-analytique d'une série trigonométrique nulle presque partout]
Nous avons montré récemment (C. R. Acad. Sci. Paris, Ser. I 336 (2003) 475–478) que la partie anti-analytique d'une série trigonométrique qui converge vers zéro presque partout peut appartenir à L2 sur le cercle. Nous montrons ici qu'elle peut même appartenir à C∞, et nous donnons le meilleur degré de régularité possible en termes de rapidité de décroissance des coefficients de Fourier. Il s'agit d'une nouvelle sorte de quasi-analyticité.
We proved recently (C. R. Acad. Sci. Paris, Ser. I 336 (2003) 475–478) that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may belong to L2 on the circle. Here we prove that it can even be C∞, and we characterize precisely the possible degree of smoothness in terms of the rate of decrease of the Fourier coefficients. This sharp condition might be viewed as a ‘new quasi-analyticity’.
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@article{CRMATH_2004__338_7_515_0,
author = {Kozma, Gady and Olevski{\i}̆, Alexander},
title = {Maximal smoothness of the anti-analytic part of a trigonometric null series},
journal = {Comptes Rendus. Math\'ematique},
pages = {515--520},
publisher = {Elsevier},
volume = {338},
number = {7},
year = {2004},
doi = {10.1016/j.crma.2004.01.025},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.01.025/}
}
TY - JOUR AU - Kozma, Gady AU - Olevskiı̆, Alexander TI - Maximal smoothness of the anti-analytic part of a trigonometric null series JO - Comptes Rendus. Mathématique PY - 2004 SP - 515 EP - 520 VL - 338 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.01.025/ DO - 10.1016/j.crma.2004.01.025 LA - en ID - CRMATH_2004__338_7_515_0 ER -
%0 Journal Article %A Kozma, Gady %A Olevskiı̆, Alexander %T Maximal smoothness of the anti-analytic part of a trigonometric null series %J Comptes Rendus. Mathématique %D 2004 %P 515-520 %V 338 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.01.025/ %R 10.1016/j.crma.2004.01.025 %G en %F CRMATH_2004__338_7_515_0
Kozma, Gady; Olevskiı̆, Alexander. Maximal smoothness of the anti-analytic part of a trigonometric null series. Comptes Rendus. Mathématique, Tome 338 (2004) no. 7, pp. 515-520. doi : 10.1016/j.crma.2004.01.025. http://www.numdam.org/articles/10.1016/j.crma.2004.01.025/
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☆ This Research was supported in part by the Israel Science Foundation.
