Cyclicity in the harmonic Dirichlet space

Abstract: The harmonic Dirichlet space $\cal{D} (\mathbb{T})$ is the Hilbert space of functions $f \in L^2(\mathbb{T})$ such that $$|f|{\cal{D} (\mathbb{T})}^2 := \sum{n\in\mathbb{Z}} (1+|n|)|\hat{f}(n)|^2 < \infty.$$ We give sufficient conditions for $f$ to be cyclic in $\cal{D} (\mathbb{T})$, in other words, for ${\zeta ^nf(\zeta):\ n\geq 0}$ to span a dense subspace of $\cal{D} (\mathbb{T})$.