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

Abstract

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 $t\ne 0$, we prove that $\{f(z^k){\}}_{k\in \mathbb{N}}$ is orthogonal in ${\mathcal{D}}_t$ only if $f=c{z}^N$ 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.

18 pages