Sampling discretization, cubature formulas and quantitative approximation in multidimensional settings

本研究的主要目的是研究采样离散化、立方公式以及它们与高维环境下定量逼近理论的关系。采样离散化是指通过固定有限点集上的函数值将连续对象转化为离散对象的过程，而立方是指通过有限点集上函数值的加权和来数值逼近函数的多维积分的方法，称为立方公式。离散化是使连续问题在计算上可行的重要步骤，而体积公式在高维积分的离散化和实用计算中起着至关重要的作用。为了确保问题的计算机可访问性，离散化的第一步通常是用低维的对应物(如多元多项式、样条、神经网络)逼近连续算子和高维函数空间的过程。在本节目中，我将重点介绍在近似和离散化过程中不可避免地出现的误差的定量估计。尽管许多对离散化至关重要的问题，但高维的立方公式和近似理论仍然悬而未决。本程序所研究的问题包括：(I)高维子空间中函数的Lq范数的抽样离散化；(Ii)高维函数空间中的偏差估计和体积公式；(Iii)能量最小化和最优体积公式之间的相互作用；(Iv)球面上局部支持的正定函数与相关区域之间的联系；(V)均匀分布点集的新构造(低偏差、体积、能量最小化、格子)；以及(Vi)高维域上逼近的无量纲估计。我的研究将使用一种新的技术，它结合了基于链和大偏差不等式的强大的概率技术，以及多维逼近理论中的各种深层结果(例如，多项式逼近中的熵数和N宽度的估计，各种多项式不等式，以及多项式逼近中的正反无量纲估计)。我也期待经典的正交多项式展开理论，特别是球谐分析，将成为我研究的有力工具。该程序涉及计算数学(即数值积分方法)、概率论、统计学、人工智能等数学领域。这一计划的科学成果预计将影响数学的几个领域，用新的结果、想法和方法丰富和交叉培养它们。

The primary goal of my research is to investigate sampling discretization, cubature formulas  and their connections with quantitative approximation theory in higher dimensional settings.  Sampling discretization refers to the process of transferring continuous objects into their discrete counterparts through function values at a fixed finite set of points, whereas cubature refers to a method for numerically approximating a multidimensional integral of a function through a  weighted sum of function values on a finite set of points, which is called a cubature formula. Discretization is an important step in making a continuous problem computationally feasible, while cubature formulas have been playing crucial roles in discretization and practical evaluation of high dimensional integrals. To ensure the problems are accessible by computers, the first step towards discretization is usually the process of approximating continuous operators and higher dimensional function spaces by lower dimensional counterparts (e.g. multivariate polynomials, splines, neural networks). In this program, I will focus on quantitative estimates of errors that inevitably arise in the process of approximation and discretization.    Many questions of utmost importance to discretization, cubature formulas and approximation theory in higher dimensions remain wide open. The questions under investigation in this program include (i)  sampling discretizations of Lq norms of functions from a high-dimensional subspace; (ii) discrepancy estimates and cubature formulas in high-dimensional function spaces; (iii) interplay between energy minimization and optimal cubature formulas; (iv) connections between locally supported positive definite functions on spheres and related domains; (v) new constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices); and (vi) dimension-free estimates for approximation on high-dimensional domains.     My research will use a new technique, which combines powerful probabilistic techniques, based on chaining and large deviation inequalities, with various deep results in multidimensional approximation theory (e.g., estimates of entropy numbers and N-widths, various polynomial inequalities,  direct and inverse dimension-free  estimates in polynomial approximation). I also expect that the theory of classical orthogonal polynomial expansions, especially spherical harmonic analysis,  will be a powerful  tool in my research.    This program is connected to computational mathematics (i.e., the methods of numerical integration), probability, statistics, artificial intelligence and other areas of mathematics. The scientific outputs of this program are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.