Power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces

Power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces $\mathcal{D}_t\ (t\in\mathbb{R})$ are treated. When $t\neq0$, it is proved that a system of functions $\{f(z^k)\}_{k\in\mathbb{N}}$ is orthogonal in $\mathcal{D}_t$ only if $f=cz^N$ for some constant $c$ and some positive integer $N$. Complete characterizations are also given of unconditional bases and frames formed by power dilation systems of Dirichlet-type spaces. Finally, these results are applied to the operator theoretic case of the moment problem on Dirichlet-type spaces.