Uniform Distribution and Harmonic Analysis

均匀分布和谐波分析

• 批准号: 1665007
• 负责人: Dmitriy Bilyk
• 金额: $ 18万
• 依托单位: University of Minnesota-Twin Cities
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1665007&HistoricalAwards=false
• 关键词: Uniform; Distribution; Harmonic; Analysis

This research project deals with the fundamental problem of approximating continuous objects by discrete ones and evaluating the errors that inevitably arise in the process. Questions of this kind come up in most branches of mathematics: probability, approximation theory, number theory, combinatorics, and discrete geometry, as well as in any discipline of science, engineering, or finance that demands computations of multivariate integrals. A number of less obvious and more profound connections have been discovered recently: some have already been formalized and understood, while others are only heuristic and await further research. Almost every pivotal development in uniform distribution theory, the focus of this project, has exploited applications of real analysis, functional analysis, and especially harmonic analysis. At the same time, numerous modern methods of harmonic analysis that were historically overlooked by experts in other fields that make use of uniform distribution theory are only starting to gain footholds in their areas. Many questions of utmost importance to uniform distribution theory, especially in higher dimensions, remain wide open. The questions under investigation in this project include the longstanding problem of precise asymptotics of optimal discrepancy in higher dimensions, one-bit compressed sensing, embeddings of metric spaces, interplay between discrepancy and energy minimization, constructions of well-distributed point sets (low-discrepancy, cubature, energy-minimizing, lattices), discrepancy estimates and numerical integration in function spaces, exploring the effect of geometry on uniform distribution properties, and problems of discrete geometry (tessellations, coverings, packings). Progress on these questions has to be tightly intertwined with advances on important problems and conjectures in analysis and other fields, gluing together a mosaic of apparently disconnected questions and topics. The outcomes of the project are expected to impact several areas of mathematics, enriching and cross-fertilizing them with new results, ideas, and methods.

本研究项目涉及用离散对象近似连续对象的基本问题，并评估在此过程中不可避免地出现的误差。这类问题出现在数学的大多数分支中：概率论、逼近论、数论、组合学和离散几何，以及任何需要计算多元积分的科学、工程或金融学科。 最近发现了许多不太明显但更深刻的联系：有些已经被形式化和理解，而另一些只是启发式的，等待进一步的研究。 几乎每一个关键的发展，在均匀分布理论，这个项目的重点，利用应用程序的真实的分析，功能分析，特别是谐波分析。与此同时，历史上被其他领域的专家忽视的许多现代谐波分析方法，利用均匀分布理论，才开始在他们的领域站稳脚跟。许多对均匀分布理论至关重要的问题，特别是在更高维度下，仍然悬而未决。该项目研究的问题包括高维最优差异的精确渐近性问题、一比特压缩感知、度量空间的嵌入、差异与能量最小化之间的相互作用、均匀分布点集的构造（低差异、体积、能量最小化、网格）、函数空间中的差异估计和数值积分，探索几何对均匀分布性质的影响，以及离散几何问题（镶嵌，覆盖，包装）。在这些问题上的进展必须与分析和其他领域的重要问题和成就的进展紧密交织在一起，把一个明显不相关的问题和主题拼凑在一起。该项目的成果预计将影响数学的几个领域，用新的结果，思想和方法丰富和交叉它们。

项目成果

The Stolarsky Principle and Energy Optimization on the Sphere

• DOI: 10.1007/s00365-017-9412-4
• 发表时间: 2018-08-01
• 期刊: CONSTRUCTIVE APPROXIMATION
• 影响因子: 2.7
• 作者: Bilyk, Dmitriy;Dai, Feng;Matzke, Ryan
• 通讯作者: Matzke, Ryan

Potential theory with multivariate kernels

多元核势理论

• DOI: 10.1007/s00209-022-03000-z
• 发表时间: 2022
• 期刊: Mathematische Zeitschrift
• 影响因子: 0.8
• 作者: Bilyk, Dmitriy;Ferizović, Damir;Glazyrin, Alexey;Matzke, Ryan W.;Park, Josiah;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr

Energy on spheres and discreteness of minimizing measures

球体上的能量和最小化措施的离散性

• DOI: 10.1016/j.jfa.2021.108995
• 发表时间: 2021
• 期刊: Journal of Functional Analysis
• 影响因子: 1.7
• 作者: Bilyk, Dmitriy;Glazyrin, Alexey;Matzke, Ryan;Park, Josiah;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr

Positive definiteness and the Stolarsky invariance principle

正定性和斯托拉斯基不变性原理

• DOI: 10.1016/j.jmaa.2022.126220
• 发表时间: 2022
• 期刊: Journal of Mathematical Analysis and Applications
• 影响因子: 1.3
• 作者: Bilyk, Dmitriy;Matzke, Ryan W.;Vlasiuk, Oleksandr
• 通讯作者: Vlasiuk, Oleksandr

Geodesic distance Riesz energy on the sphere

球体上的测地距离 Riesz 能量

• DOI: 10.1090/tran/7711
• 发表时间: 2019
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Bilyk, Dmitriy;Dai, Feng
• 通讯作者: Dai, Feng