Cyclicity in the harmonic Dirichlet space

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