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 ℓ^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 𝔻X∞ \mathbb{D}_X^\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 𝔻X∞,|ℂN=𝔻N \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = {\mathbb{D}^N} and the ℓ^p-unit balls 𝔻X∞,|ℂN=𝔹pN \mathbb{D}_X^\infty ,|{\mathbb{C}^N} = \mathbb{B}_p^N , in particular to Dirichlet-type and Dirichlet–Drury–Arveson-type spaces and algebras, as X=ℓp(ℤ+N,(1+α)s) X = {\ell ^p}\left( {_ + ^N,{{\left( {1 + \alpha } \right)}^s}} \right)) , s = (s₁, s₂, … ) and X=ℓp(ℤ+N,  (α!| α |!)t(1+| α |)s) X = {\ell ^p}\left( {\mathbb{Z}_ + ^N,\,\,{{\left( {{{\alpha !} \over {\left| \alpha \right|!}}} \right)}^t}{{\left( {1 + \left| \alpha \right|} \right)}^s}} \right) , s,t ≥ 0, as well as to their infinite variables analogues. We priviledged the largest possible scale of spaces and the most elementary instruments used.