Mathematical Sciences: Uniqueness of Multiple Trigonometric Series

数学科学：多重三角级数的唯一性

• 批准号: 9307242
• 负责人: Marshall Ash
• 金额: $ 3万
• 依托单位: DePaul University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-15 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9307242&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Uniqueness; Multiple; Trigonometric

Ash 9307242 This mathematical research focuses on problems of multiple trigonometric series. The single-variable theory of such series could be said to be reasonably well understood, although there remain a number of exceedingly difficult problems still unresolved. Multiple trigonometric series have intrinsic problems which do not allow for simple generalizations from the one-variable case. One of the primary issues of convergence results from the variety of natural ways one can try to sum a multiple series. Even the most basic question of whether a multiple series which sums to the zero function must have zero coefficients (Cantor's theorem in one variable). If unrestricted rectangular convergence is allowed then the answer is affirmative in all dimensions. This was only proved two years ago. For spherical and square convergence, the answer is still unknown, except for spherical in dimension two. Work will be done to extend the recent results obtained for rectangular convergence to the remaining cases. Related work will be carried out on the almost everywhere convergence of multiple Fourier series. For functions in the Lebesgue spaces with power less than two, spherical partial sums may diverge on sets of positive measure. Nothing is known about convergence for functions with finite quadratic norm (Hilbert space). To study this class, work will first concentrate on radial functions to see if counterexamples already exist within this group. The influence of trigonometric series on the development of modern mathematical analysis is impossible to record in this short space. The basic U.S. graduate courses are replete with byproducts of the research efforts that have gone into our attempts to understand these building blocks of mathematical synthesis used universally by all scientists. Yet the underlying summability properties of the series remains, to a large extent, an enigma. This project seeks to build on a remarkable breakthrough which occurred when four researchers settled one of the most famous unsolved problems of multiple Fourier series two years ago. ***

灰9307242 数学研究的重点是多重三角级数问题。 这种级数的单变量理论可以说是相当好理解的，尽管仍然有一些非常困难的问题尚未解决。 多个三角级数有内在的问题，不允许简单的推广从一个变量的情况。 收敛性的主要问题之一来自于人们可以尝试对多重级数求和的各种自然方法。 即使是最基本的问题，是否一个多重级数的总和为零的功能必须有零系数（康托定理在一个变量）。 如果允许无限制的矩形收敛，那么在所有维度上答案都是肯定的。 这一点在两年前才得到证实。 对于球面收敛和平方收敛，答案仍然未知，除了二维球面收敛之外。 工作将做矩形收敛到其余的情况下，最近获得的结果。对多重Fourier级数的几乎处处收敛性进行了相关的研究。 对于Lebesgue空间中幂小于2的函数，球面部分和可以在正测度集上发散。 对于具有有限二次范数的函数（希尔伯特空间）的收敛性一无所知。 为了研究这个类，工作将首先集中在径向函数，看看反例是否已经存在于这个组。 三角级数对现代数学分析发展的影响是不可能在这短短的篇幅中记载的。 基本的美国研究生课程充满了研究工作的副产品，这些研究工作已经进入我们试图理解所有科学家普遍使用的数学综合的基石。 然而，在很大程度上，级数的基本可求和性质仍然是一个谜。 该项目旨在建立在两年前四名研究人员解决了多个傅里叶级数中最著名的未解决问题之一时取得的显着突破的基础上。 ***