Cyclicity in the harmonic Dirichlet space

TL;DR

This paper investigates the conditions under which functions in the harmonic Dirichlet space are cyclic, meaning their polynomial multiples densely span the space, contributing to the understanding of cyclic vectors in harmonic analysis.

Contribution

It provides sufficient conditions for functions to be cyclic in the harmonic Dirichlet space, advancing the theory of cyclic vectors in this functional setting.

Findings

Identifies conditions for cyclicity in the harmonic Dirichlet space

Establishes criteria for polynomial multiples to be dense

Enhances understanding of harmonic Dirichlet space structure

Abstract

The harmonic Dirichlet space $\cal{D} (\mathbb{T})$ is the Hilbert space of functions $f \in L^2(\mathbb{T})$ such that $$\|f\|_{\cal{D} (\mathbb{T})}^2 := \sum_{n\in\mathbb{Z}} (1+|n|)|\hat{f}(n)|^2 < \infty.$$ We give sufficient conditions for $f$ to be cyclic in $\cal{D} (\mathbb{T})$, in other words, for $\{\zeta ^nf(\zeta):\ n\geq 0\}$ to span a dense subspace of $\cal{D} (\mathbb{T})$.