Approximation, Equilibrium Measures and Discrepancy over Domains, Finite Fields and Smooth Manifolds

域、有限域和光滑流形上的近似、平衡测度和差异

• 批准号: 0555839
• 负责人: Steven Damelin
• 金额: --
• 依托单位: Georgia Southern University Research and Service Foundation, Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0555839&HistoricalAwards=false
• 关键词: Approximation; Equilibrium; Measures; Discrepancy; over

ABSTRACTThe research will primarily focus on three areas: discrepancy estimates over domains and manifolds for classes of smooth and singular functions and kernels which depend only on distances between points in Euclidean space; the study of low discrepancy sequences such as configurations interacting via a pairwise repulsive interaction on a fixed manifold and bases of linear independent vectors over a finite field; the study of tools for good approximation of projective, group invariant and singular operators on domains and smooth manifolds. The research is expected to develop integration estimates over compact homogenous manifolds using energy functions and zonal kernels; produce theorems on separation and mesh norm properties of Riesz configurations as well as exact formulas for maximal independent sets of vectors over finite fields of fixed order; produce theorems on hyperinterpolation operators and supports of equilibrium measures. In most phases of the research, approximation theory and potential theory are expected to be useful tools.The research will primarily focus on three areas: (i) the approximation of multidimensional integrals and the study of functions which describe similarities in a finite set of data; (ii)the study of minimal energy (ground state) particles interacting via a repulsive force and linear codes with certain parameters; (iii) the study of a broad range of tools which arise in approximation theory and orthogonal polynomials. Regarding (i), our goal is to develop approximation estimates of multidimensional integrals using distance only depending functions. The former arise in the analysis of satellite data on the surface of the earth and in mathematical finance while the later occur in meaningful structures and descriptions of large data sets for example in imaging and wireless networks. Regarding (ii), our goal is to study clustering properties of minimal energy particles which are useful in understanding best packing and in the understanding of the physics of self-assembling materials. We also expect to prove the existence of linear codes which areuseful for the problem of transmitting information effectively and accurately. Regarding (iii), our goal is to understand interpolation operators on manifolds as a means to solving differential equations and supports of minimizers on the circle which are used in random matrix theory in mathematical physics.

本文的研究将主要集中在三个方面：仅依赖于欧氏空间中点之间距离的光滑和奇异函数和核在区域和流形上的偏差估计；低偏差序列的研究，例如通过固定流形和有限域上线性独立向量基的成对排斥相互作用而相互作用的构形；区域和光滑流形上射影、群不变和奇异算子良好逼近的工具的研究。这项研究有望利用能量函数和带状核在紧致齐次流形上发展积分估计；产生关于Riesz构型的分离性和网格范数性质的定理以及关于定阶有限域上的极大独立向量集的精确公式；产生关于超插值算子的定理和平衡测度的支撑。在研究的大多数阶段，近似理论和位势理论有望成为有用的工具。研究将主要集中在三个方面：(I)多维积分的近似和描述有限数据集中相似性的函数的研究；(Ii)最小能量(基态)粒子通过排斥力和具有某些参数的线性代码相互作用的研究；(Iii)在近似理论和正交多项式中出现的广泛工具的研究。关于(I)，我们的目标是利用仅依赖于距离的函数来发展多维积分的近似估计。前者出现在对地球表面卫星数据的分析和数学金融学中，而后者出现在大型数据集的有意义的结构和描述中，例如在成像和无线网络中。关于(Ii)，我们的目标是研究最低能量粒子的团簇性质，这对于理解最佳堆积和理解自组装材料的物理是有用的。我们还希望证明线性码的存在，这对于有效和准确地传输信息是有用的。关于(III)，我们的目标是理解流形上的内插算子作为一种求解微分方程的手段和圆周上极小点的支持，这在数学物理中的随机矩阵理论中使用。