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

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 ${\mathrm{\ell }}^p$ 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 ${\mathbb{D}}_{X^{\mathrm{\prime }}}^{\mathrm{\infty }}$. 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 ${\mathbb{D}}_{X^{\mathrm{\prime }}}^{\mathrm{\infty }}\mathrm{\mid }{\mathbb{C}}^N={\mathbb{D}}^N$ and the ${\mathrm{\ell }}^p$-unit balls ${\mathbb{D}}_{X^{\mathrm{\prime }}}^{\mathrm{\infty }}\mathrm{\mid }{\mathbb{C}}^N={\mathbb{B}}_p^N$, in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as $X={\mathrm{\ell }}^p({\mathbb{Z}}_{+}^N,(1+\alpha {)}^s)$, $s=(s_1,s_2,\dots )$ and $X={\mathrm{\ell }}^p({\mathbb{Z}}_{+}^N,(\frac{\alpha !}{\mathrm{\mid }\alpha \mathrm{\mid }!}{)}^t(1+\mathrm{\mid }\alpha \mathrm{\mid }{)}^s)$, $s,t\ge 0$, as well as to their infinite variables analogues. We privileged the largest possible scale of spaces and the most elementary instruments used.