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'.
|Translated title of the contribution||Maximal smoothness of the anti-analytic part of a trigonometric null series|
|Number of pages||6|
|Journal||Comptes Rendus Mathematique|
|State||Published - 1 Apr 2004|