Comptes Rendus
no. 7
Mathematical Analysis/Harmonic Analysis
Maximal smoothness of the anti-analytic part of a trigonometric null series
[Régularité maximale de la partie anti-analytique d'une série trigonométrique nulle presque partout]

Présenté par : Jean-Pierre Kahane

Gady Kozma1 ; Alexander Olevskiı̆1
1 Tel Aviv University, School of Mathematics, Ramat Aviv 69978, Israel
Comptes Rendus. Mathématique, Volume 338 (2004) no. 7, pp. 515-520.

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’.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.01.025
Gady Kozma 1 ; Alexander Olevskiı̆ 1

1 Tel Aviv University, School of Mathematics, Ramat Aviv 69978, Israel 
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Gady Kozma; Alexander Olevskiı̆. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.01.025/

[1] N.K. Bary A Treatise on Trigonometric Series, Pergamon Press, 1964

[2] A. Beurling Quasianaliticity, The Collected Works, vol. 1, Birkhaüser, 1989 (pp. 396–431)

[3] A.A. Borichev; A.A. Borichev Boundary uniqueness theorems for almost analytic functions, and asymmetric algebras of sequences, Mat. Sb., Volume 136 (1988) no. 2, pp. 324-340 (in Russian) English translation Math. USSR-Sb., 64, 1989, pp. 323-338

[4] J.P. Kahane; R. Salem Ensemles parfaits et series trigonometriques, Hermann, 1994

[5] A. Kechris; A. Louveau Descriptive Set Theory and the Structure of Sets of Uniqueness, Cambridge University Press, 1987

[6] G. Kozma; A. Olevskiı̆ A null series with small anti-analytic part, C. R. Acad. Sci. Paris, Ser. I, Volume 336 (2003), pp. 475-478

[7] S. Mandelbrojt Séries de Fourier et classes quasi-analytiques, de Fonctions, Gauthier-Villard, Paris, 1935

Cité par Sources :

☆ This Research was supported in part by the Israel Science Foundation.

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