Cyclicity in the harmonic Dirichlet space

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 $\{ζ^nf(ζ):\ n\geq 0\}$ to span a dense subspace of $\cal{D} (\mathbb{T})$.

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