Uniqueness theorems for Walsh series under a strong condition
The author examines sets of uniqueness for the class of Walsh series which satisfy \[ \sum^{2^{n-1}-1}_{k= 2^ n} a_ k w_ k(x)= o(1) \] uniformly in \(x\) as \(n\to \infty\). He proves a structural theorem (which shows that every \(H^{(1)}\)-set on the dyadic group is a set of uniqueness for this class), and proves that some sets of Haar measure zero and all sets of positive Haar measure on the dyadic group are sets of multiplicity (i.e., not sets of uniqueness) for this class.
scientific article; zbMATH DE number 633877