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

Power dilation systems { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} in Dirichlet-type spaces D t   ( t ∈ R ) \mathcal {D}_t\ (t\in \mathbb {R}) are treated. When t ≠ 0 t\neq 0 , it is proved that a system of functions { f ( z k ) } k ∈ N \{f(z^k)\}_{k\in \mathbb {N}} is orthogonal in D t \mathcal {D}_t only if f = c z N f=cz^N for some constant c c and some positive integer N 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.