Cyclicity in the harmonic Dirichlet space

Abstract

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

Cited by in corpus (1)

• Cyclicity in ${\ell }^p$ spaces and zero sets of the Fourier transforms