Sets of uniqueness for a certain class ℳ_{𝜖} on the dyadic group

Abstract

For each sequence ε = { ε n } \varepsilon = \left \{ {{\varepsilon _n}} \right \} of real numbers which satisfies lim inf n → ∞ ε 2 n + 1 / ε 2 n > 0 \lim {\inf _{n \to \infty }}{\varepsilon _{{2^{n + 1}}}}/{\varepsilon _{{2^n}}} > 0 and ε n ↓ 0 {\varepsilon _n} \downarrow 0 as n → ∞ n \to \infty , let M ε {\mathfrak {M}_\varepsilon } denote the set of all Walsh series μ ∼ ∑ k = 0 ∞ μ ^ ( k ) w k ( x ) \mu \sim \sum \nolimits _{k = 0}^\infty {\hat \mu (k){w_k}(x)} such that ∑ k = 0 ∞ ε k | μ ^ ( k ) | 2 > ∞ \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 M ε {\mathfrak {M}_\varepsilon } .