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Analytic representation
Đề tài " Analytic representation of functions and a new quasianalyticity threshold " We characterize precisely the possible rate of decay of the anti-analytic half of a trigonometric series converging to zero almost everywhere. 1. Introduction 1.1. In 1916, D. E. Menshov constructed an example of a nontrivial trigonometric series on the circle T
∞
(1)
n=−∞
c(n)eint
which converges to zero almost everywhere (a.e.). Such series are called nullseries. This result was the origin of the modern theory of uniqueness in Fourier analysis, see [Z59], [B64], [KL87], [KS94]. Clearly for such a series |c(n)|2 = ∞. A less trivial observation is that a null series cannot be analytic, that is, involve positive frequencies only. ...
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