Operator Theory and Quadrature Formulas

算子理论和求积公式

• 批准号: 9800666
• 负责人: Mihai Putinar
• 金额: --
• 依托单位: University of California-Santa Barbara
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9800666&HistoricalAwards=false
• 关键词: Operator; Theory; Quadrature; Formulas

Principal Investigator: Mihai Putinar The project focuses on three themes, all well connected to each other: quadrature domains for harmonic functions via hyponormal operators of rank-one self-commutator, cubature formulas in several variables for polynomials of a fixed degree via dilation theory (with prescribed commutators, for tuples of self-adjoint operators) and the spectral picture of spherical isometries in a Hilbert space. The basis for the first subject is the progress made in the last five years in the study of quadrature domains with linear algebra techniques, and specifically using determinantal functions of pairs of matrices. There are many analogies with classical perturbation theory of self-adjoint operators which point to several open problems resulting from this approach. The second theme is based on the observation made a couple of years ago by the P.I. that practically all facts connected with multivariate cubatures can be translated into familiar terms of dilation theory (of commuting Jacobi matrices). It is expected that this point of view will be successfully exploited by both sides. The third subject is a specialization of the second, for positive measures supported by odd spheres. The complex variables and the associated Toeplitz operators (which are typical examples of spherical isometries-- ubiquitous objects in modern operator theory) are expected to carry in a closed, flexible form the basic information about the original measure. Inverse problems, that is the reconstruction of complex objects from partial data, are everywhere present in modern science and technology. To give a few classical examples we can mention the vibrations of a string, the shape of a planet knowing its distant gravitation field, or the evolution of an oil spill in water. The proposal is aimed at solving some mathematical questions related to inverse problems related to shapes or to mass distributions. A history of more than one century of remarkable discoveries in this area will be comb ined with recent advances in modern mathematics. The resulting work will be significant for the present research frontier in mathematics and potentially for applied fields such as geophysics, image processing or tomography.

主要研究者：Mihai Putinar该项目侧重于三个主题，所有这些主题都相互关联：通过秩1自交换子的亚正规算子的调和函数的求积域，通过膨胀理论（具有规定的交换子，用于自伴算子元组）的固定次数多项式的多变量体积公式，以及希尔伯特空间中球面等距的谱图。第一个主题的基础是过去五年来利用线性代数技术研究求积域所取得的进展，特别是使用矩阵对的行列式函数。有许多类似的经典扰动理论的自伴运营商指出几个开放的问题，从这种方法。第二个主题是基于几年前私家侦探的观察。实际上，所有与多元立方有关的事实都可以转化为（交换雅可比矩阵的）扩张理论的熟悉术语。预计双方将成功地利用这一观点。第三个主题是第二个主题的专门化，用于由奇数领域支持的积极措施。复变量和相关的Toeplitz算子（它们是球面等距的典型例子--现代算子理论中无处不在的对象）被期望以封闭的、灵活的形式携带关于原始测度的基本信息。反问题，即由部分数据重建复杂对象，在现代科学技术中无处不在。举几个经典的例子，我们可以提到弦的振动，知道其遥远引力场的行星的形状，或者水中石油泄漏的演变。该建议的目的是解决一些数学问题的逆问题有关的形状或质量分布。一个多世纪的历史显着的发现在这一领域将结合在现代数学的最新进展。由此产生的工作将是显着的，目前的研究前沿数学和潜在的应用领域，如电子物理学，图像处理或断层扫描。