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

本文的研究主要集中在三个方面：一类光滑奇异函数和核在区域和流形上的差异估计，低差异序列的研究，如固定流形上通过两两相斥相互作用的构型和有限域上线性独立向量基的研究;研究在域和光滑流形上的射影、群不变和奇异算子的良好逼近的工具。该研究预计将开发使用能量函数和带状核在紧致齐次流形上的积分估计;产生Riesz配置的分离和网格范数性质的定理以及固定阶有限域上的最大独立向量集的精确公式;产生超插值算子和平衡测度的支持的定理。在研究的大部分阶段，近似理论和位势理论都是有用的工具，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相似性的函数的研究;（ii）最小能量的研究（基态）粒子通过排斥力和线性代码与某些参数相互作用;（iii）研究近似理论和正交多项式中出现的各种工具。关于（i），我们的目标是使用仅依赖于距离的函数来开发多维积分的近似估计。前者出现在地球表面的卫星数据分析和数学金融中，而后者出现在有意义的结构和大型数据集的描述中，例如成像和无线网络。关于（ii），我们的目标是研究最小能量粒子的聚集特性，这对于理解最佳堆积和理解自组装材料的物理学是有用的。我们还期望证明线性码的存在性，这对于有效和准确地传递信息的问题是有用的。关于（iii），我们的目标是理解流形上的插值算子作为求解微分方程的一种手段，以及在数学物理中的随机矩阵理论中使用的圆上极小元的支持。

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.