Mathematical Sciences: Function and Operator Theory on Holomorphic Spaces

数学科学：全纯空间上的函数和算子理论

• 批准号: 9622890
• 负责人: Zhijian Wu
• 金额: $ 4.73万
• 依托单位: University of Alabama Tuscaloosa
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622890&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Function; Operator; Theory

DMS-9622890 PI: Zhijian Wu University of Alabama @ Tuscaloosa In the past several years, the principal investigator has mainly focused his research on function and operator theory on analytic (one or several variables) or monogenic function spaces. The operators include Hankel, Toeplitz, commutators, multipliers, bilinear forms, approximation and atomic decomposition. The spaces (analytic or harmonic) are Bergman, Hardy, Dirichlet and the scale of capacity related spaces which includes the Bloch space and the space of bounded mean oscillation functions. The study also involves differential operators such as d-bar, Laplacian, Div, Curl and Dirac operators. The principal investigator proposes to continue his ongoing study, to explore more connections and applications of function and operator theory to other areas. Although there are many problems related to the study which arise in nature, the principal investigator plans to focus specifically on the aforementioned operators on potential related function spaces, the function theory of these spaces, Nehari type problems in Dirichlet spaces, minimum solution operators to some differential systems and Clifford analysis in the theory of compensated compactness. He will use function and operator theoretical tools and techniques in harmonic analysis, together with new ideas and methods in his investigation. Beside their own charm in pure mathematics, problems in this project have rich connections and applications to many fields of applied sciences. For example Hankel operators (or more generally commutators and bilinear forms) are very popular in various aspects of engineering design and system science. Their behavior provides important information to automatic control, navigation systems and robust stabilization. The study of Hankel and related operators in various underlying spaces will provide effective ways to deal with control problems in different situations. Another good example is the recent discover o f the connection between Clifford analysis and the compensated compactness phenomenon in non-linear partial differential equations of multi-variable spaces. Experience and previous achievement indicate that the principal investigator has the potential to accomplish his goal.

DMS-9622890 PI：Zhijian Wu亚拉巴马大学@塔斯卡卢萨 在过去的几年里，主要研究者主要集中在他的研究功能和算子理论的分析（一个或多个变量）或单演函数空间。运营商包括汉克尔， Toeplitz，乘法器，乘法器，双线性形式，近似和原子分解。空间（解析或调和）是Bergman，哈代，Dirichlet和容量相关空间的标度，包括Bloch空间和有界平均振动函数空间。该研究还涉及微分算子，如d-bar，Laplacian，Div，Curl和Dirac算子。 首席研究员建议继续他正在进行的研究，探索更多的连接和应用功能和算子理论到其他领域。虽然有许多问题与研究中出现的性质，主要研究人员计划专门集中在上述运营商的潜在相关的功能空间，这些空间的函数理论，Nehari型问题在狄利克雷空间，最小解运营商的一些微分系统和Clifford分析理论的补偿紧。他将使用函数和运算符 谐波分析的理论工具和技术，以及新的 在他的调查中的思想和方法。 这类问题除了具有纯数学的魅力外，还与应用科学的许多领域有着密切的联系和应用。例如，汉克尔算子（或更一般的双线性形式）在工程设计和系统科学的各个方面都非常流行。它们的行为为自动控制、导航等提供了重要的信息 系统和鲁棒稳定。Hankel算子及相关算子的研究 将提供有效的方法来处理控制 不同情况下的问题。另一个很好的例子是最近发现的 Clifford分析与补偿紧性的关系 多元非线性偏微分方程中的一个现象 空间.经验和以往的成就表明， 调查员有可能实现他的目标。