On the Cyclicity of Dilated Systems in Lattices: Multiplicative Sequences, Polynomials, Dirichlet-type Spaces and Algebras

TL;DR

This paper investigates the cyclicity of dilated systems within various lattice-based function spaces, including weighted $ ext{ell}^p$ spaces, focusing on multiplicative sequences, polynomials, and Dirichlet-type spaces, with applications to infinite-dimensional domains.

Contribution

It extends the analysis of cyclicity to general lattice frameworks and specific function spaces like Dirichlet and Dirichlet--Drury--Arveson spaces, using elementary methods.

Findings

Cyclicity characterized in weighted $ ext{ell}^p$ spaces.

Analysis of multiplicative and completely multiplicative sequences.

Results applicable to infinite-dimensional Reinhardt domains.

Abstract

The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice $X$, including weighted $\ell^p$ spaces. In particular, general multiplicative and completely multiplicative sequences are treated. After the Fourier--Bohr transformation, we deal with the cyclicity property in function spaces on the corresponding infinite dimensional Reinhardt domain $\mathbb{D}^\infty_{X'}$. Functions with (weakly) dominating free term and (in particular) linearly factorable functions are considered. The most attention is paid to the cases of the polydiscs $\mathbb{D}^\infty_{X'}|\mathbb{C}^N=\mathbb{D}^N$ and the $\ell^p$-unit balls $\mathbb{D}^\infty_{X'}|\mathbb{C}^N=\mathbb{B}_p^N$, in particular to Dirichlet-type and Dirichlet--Drury--Arveson-type spaces and algebras, as $X=\ell^p(\mathbb{Z}_+^N,(1+\alpha)^s)$, $s=(s_1,s_2,\dots)$ and $X=\ell^p(\mathbb{Z}_+^N,(\frac{\alpha!}{|\alpha|!})^t(1+|\alpha|)^s)$, $s,t\geq 0$, as well as to their infinite variables analogues. We privileged the largest possible scale of spaces and the most elementary instruments used.