Mathematical Analysis/Harmonic Analysis
Maximal smoothness of the anti-analytic part of a trigonometric null series
Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 515-520.

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’.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2004.01.025
Kozma, Gady 1; Olevskiı̆, Alexander 1

1 Tel Aviv University, School of Mathematics, Ramat Aviv 69978, Israel
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Kozma, Gady; Olevskiı̆, Alexander. Maximal smoothness of the anti-analytic part of a trigonometric null series. Comptes Rendus. Mathématique, Volume 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.