Invariant subspaces of the harmonic Dirichlet space with large co-dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift $f \to \zeta f$) of the harmonic Dirichlet space $D$. Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces ${\mathcal F} \subset D$ with $\mbox {dim}({\mathcal F}/ \zeta {\mathcal F}) = n$, $n \in % {\mathbb N} \cup \{\infty \}$. We will also generalize this to the Dirichlet classes $D_{\alpha }$, $0 < \alpha < \infty$, as well as the Besov classes $B^{\alpha }_{p}$, $1 < p < \infty$, $0 < \alpha < 1$.