Uniform Distribution and Harmonic Analysis

均匀分布和谐波分析

基本信息

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

项目摘要

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.

该研究项目涉及用离散对象逼近连续对象并评估该过程中不可避免地出现的误差的基本问题。此类问题出现在大多数数学分支中：概率、逼近论、数论、组合学和离散几何，以及任何需要多元积分计算的科学、工程或金融学科。最近发现了一些不太明显和更深刻的联系：一些已经被形式化并被理解，而另一些只是启发式的，有待进一步研究。均匀分布理论的几乎每一个关键发展（该项目的重点）都开发了实分析、泛函分析，尤其是调和分析的应用。与此同时，许多利用均匀分布理论的现代调和分析方法在历史上被其他领域的专家所忽视，现在才刚刚开始在各自的领域站稳脚跟。许多对均匀分布理论至关重要的问题，尤其是在更高维度上，仍然悬而未决。该项目正在研究的问题包括长期存在的高维最优差异的精确渐近问题、一位压缩感知、度量空间的嵌入、差异和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，格子），函数空间中的差异估计和数值积分，探索几何对均匀分布特性的影响，以及离散几何问题（镶嵌、覆盖物、包装）。这些问题的进展必须与分析和其他领域的重要问题和猜想的进展紧密地交织在一起，将看似互不相关的问题和主题粘合在一起。该项目的成果预计将影响数学的多个领域，通过新的结果、想法和方法丰富和交叉促进它们。

项目成果

期刊论文数量（10）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

The Stolarsky Principle and Energy Optimization on the Sphere

斯托拉斯基原理和球体能量优化

DOI：
10.1007/s00365-017-9412-4
发表时间：
2018-08
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：
2022-07
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：
2021-06
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

General and refined Montgomery Lemmata

一般和精炼的蒙哥马利引理

DOI：
10.1007/s00208-018-1738-0
发表时间：
2018-01-23
期刊：
Mathematische Annalen
影响因子：
1.4
作者：
D. Bilyk;F. Dai;S. Steinerberger
通讯作者：
S. Steinerberger

Optimal measures for $p$-frame energies on spheres

球体上 $p$ 框架能量的最佳测量

DOI：
10.4171/rmi/1329
发表时间：
2022-01
期刊：
Revista Matemática Iberoamericana
影响因子：
0
作者：
Bilyk, Dmitriy;Glazyrin, Alexey;Matzke, Ryan;Park, Josiah;Vlasiuk, Oleksandr
通讯作者：
Vlasiuk, Oleksandr