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

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 $X=\ell^p(\mathbb{Z}_+^N,(1+α)^s)$, $s=(s_1,s_2,\dots)$ and $X=\ell^p(\mathbb{Z}_+^N,(\frac{α!}{|α|!})^t(1+|α|)^s)$, $s,t\geq 0$, as well as to their infinite variables analogues. We privileged the largest possible scale of spaces and the most elementary instruments used.