Mathematical Sciences: Uniqueness of Multiple Trigonometric Series

本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容易理解的，尽管仍有许多极其困难的问题尚未解决。多重三角级数有其固有的问题，不能从单变量的情况进行简单的推广。收敛性的主要问题之一来自于对多个级数求和的各种自然方法。即使是最基本的问题，一个和为零函数的多重级数是否必须有零系数（单变量康托尔定理）。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。这在两年前才得到证实。对于球面和平方收敛，答案仍然是未知的，除了球面在二维。我们将把最近得到的矩形收敛的结果推广到其他情况。本文将对多重傅立叶级数的几乎处处收敛性进行相关研究。对于幂小于2的Lebesgue空间中的函数，球面部分和可以在正测度集合上发散。对于有限二次范数函数（希尔伯特空间）的收敛性，我们一无所知。为了学习这个类，工作将首先集中在径向函数上，看看在这个组中是否已经存在反例。三角级数对现代数学分析发展的影响是不可能在这么短的篇幅内记录下来的。美国研究生的基础课程充满了研究努力的副产品，这些研究努力是为了理解所有科学家普遍使用的数学综合的组成部分。然而，在很大程度上，该级数的潜在可求和性仍然是一个谜。该项目旨在建立在一个显著的突破，当四名研究人员解决了两年前最著名的多次傅立叶级数未解决的问题之一。＊＊＊

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. ***