Maximal smoothness of the anti-analytic part of a trigonometric null series

Details Maximal smoothness of the anti-analytic part of a trigonometric null series Maximal smoothness of the anti-analytic part of a trigonometric null series

2005-10-19

arxiv.org/abs/math/0510401v1

C. R. Math. Acad. Sci. Paris 338:7 (2004), 515-520

Mathematics

Classical Analysis and ODEs

6 pages. Announcement of math.CA/0406261 and sketch of the proof of one half of it. Should be identical to journal version exc

Scientific paper

10.1016/j.crma.2004.01.025

We proved recently math.CA/0510403 that the anti-analytic part of a trigonometric series, converging to zero almost everywhere, may be square integrable on the circle. Here we prove that it can even be infinitely differentiable, 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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