Dirichlet sets and some uniqueness theorems for Walsh series
\textit{J. P. Kahane} [J. Approximation Theory 2, 233-236 (1969; Zbl 0175.352)] introduced the concept of strong U-set and proved that a Dirichlet set is a strong U-set for trigonometric series. The author [Can. J. Math. 34, 759-764 (1982; Zbl 0464.43005)] proved that a Dirichlet set is a U- set for Walsh series. The following results hold; (1) If \(\Delta\) is a subgroup of non-negative integers (as a subgroup of the dyadic field), then \(H(\Delta)=\{x:\) \(w_ k(x)=1\) for all \(k\in \Delta \}\) is a Dirichlet set in the dyadic group. (2) A Walsh series \(\mu \equiv \sum^{\infty}_{k=0}{\hat \mu}(k)w_ k(x)\) satisfies \(\lim_{n\to \infty}\sum^{2^ n-1}_{k=0}{\hat \mu}(k) w_ k(x)=0\) for \(x\not\in H(\Delta)\) if and only if \({\hat \mu}(n\dot +k)={\hat \mu}(n9\) for all \(k\in \Delta\) and \(n=0,1,....\) (3) A Dirichlet set in the dyadic group is a strong U-set for Walsh series.
scientific article; zbMATH DE number 3973530