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

In this paper, we concentrate on power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces $\mathcal{D}_t\ (t\in\mathbb{R})$. When $t\neq0$, we prove that $\{f(z^k)\}_{k\in\mathbb{N}}$ is orthogonal in $\mathcal{D}_t$ only if $f=cz^N$ for some constant $c$ and some positive integer $N$. We also give complete characterizations of unconditional bases and frames formed by power dilation systems for Drichlet-type spaces.