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

Abstract

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 ℓp 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 ℓp-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.