Dirichlet 级数、Bohr变换和无限个变量的Hilbert模

Dirichlet级数空间, Bohr变换与无限个变量的Hardy空间的研究为Riemann假设, Beurling的膨胀系完全性问题、无限个变量函数空间上的算子理论和算子代数的研究开辟了广阔的视野, 是目前泛函分析、算子论、调和分析分析以及函数论等多个数学分支的重要交叉领域。本项目将结合多圆盘Hardy空间的函数论和“硬”调和分析, 发展可数无限个变量的解析函数空间上的函数论、调和分析及其相关的算子理论。结合有限维区域上的解析Hilbert模理论和无限变量的交换代数, 研究无限个变量Hardy子模的分类及其循环向量，进而对Beurling的膨胀系完全性问题，Riesz基问题、正交系问题、Dirichlet级数函数空间的不变子空间问题和Dirichlet级数零点的分布等基本的空间理论问题、算子论问题和零点分布密度等核心问题的研究作出贡献。

The study of the space of Dirichlet series, the Bohr transform and the Hardy space of infinitely many variables broadens our scope of research for Riemann Hypothesis, Beurling Completeness Problem of dilation system, and the Operator Theory and Operator Algebra over function spaces of infinitely many variables. The topic has become an important cross field between Functional Analysis, Operator Theory, Harmonic analysis and Function Theory. The present program will make use of function theory on the Hardy space over polydiscs and solid harmonic analysis, to develop function theory, harmonic analysis and related operator theory on analytic function space of infinitely many variables. We will combine Analytic Hilbert Module Theory over finite-dimensional domains and Commutative Algebra of infinitely many variables, to investigate cyclic vectors and classification of Hardy submodules of infinitely many variables. This program will make valuable contribution for the study on some central problems such as Beurling’s completeness problem of dilation systems, Riesz’s basis problem, orthogonal system problem, invariant subspace problem on function spaces in infinitely many variables, and the zero point distribution problem of Dirichlet series.