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

TL;DR

This paper investigates power dilation systems in Dirichlet-type spaces, establishing orthogonality conditions and characterizations of bases and frames, thereby advancing understanding of their structure in functional analysis.

Contribution

It provides new characterizations of orthogonality, bases, and frames for power dilation systems in Dirichlet-type spaces, especially for the case when t ≠ 0.

Findings

Power dilation systems are orthogonal only for monomials in Dirichlet-type spaces when t ≠ 0.

Complete characterizations of unconditional bases in these spaces.

Complete characterizations of frames formed by power dilation systems.

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 $t\neq0$, 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.