Uniqueness for trigonometric series
A question going back to N. K. Bary in 1923 asked for conditions on a sequence \((a_ n)\) which would ensure that, if \(\sum_{| n| \leq N}a_ ne^{int}\to 0\) a.e. as \(N\to \infty\), then \(a_ n=0\) for all n. A sufficient condition, for example (following from Carleson's Theorem but also susceptible to a more elementary proof), is that \(\sum^{\infty}_{-\infty}| a_ n|^ 2<\infty\). Another sufficient condition is that \(| a_ n| \leq \phi (n)\), where \(\phi (\pm n_ j)=1\) on some lacunary sequence \((n_ j)\), with \(\phi (n)=0\) otherwise. Letting \(\phi (n)>0(n=0,1,...)\) with \(\sum^{\infty}_{0}\phi (n)^ 2=+\infty\), the following long-standing conjectures are considered: (\(\alpha)\) \(\exists a_ n\), \(| a_ n| \leq \phi (| n|),\quad \sum_{| n| \leq N}a_ ne^{int}\to 0\) a.e. (N\(\to \infty)\), but not all the \(a_ n\) are zero; (\(\beta)\) \(\exists\) a non-zero measure \(\mu\) whose closed support has Lebesgue measure zero yet \(| {\hat \mu}(n)| \leq \phi (| n|)\) for all n; (\(\gamma)\) \(\exists\) a non-zero singular measure \(\mu\) with \(| {\hat \mu}(n)| \leq \phi (| n|)\) for all n. The author has developed and extended seminal work by \textit{O. S. Ivašev-Musatov} [Izv. Akad. Nauk SSSR, Ser. Mat. 21, 559-578 (1957; Zbl 0082.280)]; in particular, he has shown elsewhere [Lect. Notes Math. 781, 102-120 (1980; Zbl 0452.42007)] that conjecture (\(\beta)\) is false. In the present paper, which contains on the way a number of interesting and powerful results on trigonometric series with slowly decreasing coefficients, it is proved that conjectures (\(\alpha)\) and (\(\gamma)\) are true.
scientific article; zbMATH DE number 4075742