Sets of uniqueness for a certain class $\mathcal {M}_\varepsilon$ on the dyadic group

Abstract:

For each sequence $\varepsilon = \left \{ {{\varepsilon _n}} \right \}$ of real numbers which satisfies $\lim {\inf _{n \to \infty }}{\varepsilon _{{2^{n + 1}}}}/{\varepsilon _{{2^n}}} > 0$ and ${\varepsilon _n} \downarrow 0$ as $n \to \infty$, let ${\mathfrak {M}_\varepsilon }$ denote the set of all Walsh series $\mu \sim \sum \nolimits _{k = 0}^\infty {\hat \mu (k){w_k}(x)}$ such that $\sum \nolimits _{k = 0}^\infty {{\varepsilon _k}{{\left | {\hat \mu (k)} \right |}^2} < \infty }$. We give a necessary and sufficient condition for a subset of the dyadic group to be a set of uniqueness for ${\mathfrak {M}_\varepsilon }$.