Mathematical Sciences: Uniqueness of Multiple Trigonometric Series

ASH 9307242这项数学研究集中在多重三角级数的问题上。这种级数的单变量理论可以说是相当好地被理解了，尽管仍然有一些非常困难的问题仍然没有解决。多重三角级数有固有的问题，不允许从单变量的情况进行简单的推广。收敛的主要问题之一是人们可以尝试对多个级数求和的各种自然方法。即使是一个最基本的问题，即求和为零的重级数是否必须有零系数(康托一元定理)。如果允许无限制的矩形收敛，则所有维度的答案都是肯定的。这一点在两年前才得到证实。对于球面和平方收敛，答案仍然未知，除了二维的球面。将开展工作，将矩形收敛的最新结果推广到其余情况。关于多重傅里叶级数的几乎处处收敛的相关工作也将展开。对于幂小于2的勒贝格空间中的函数，球面部分和可能在正测度集上发散。关于具有有限二次范数的函数(Hilbert空间)的收敛，我们一无所知。为了学习这门课，我们将首先集中学习径向函数，看看这门课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，是不可能在这短短的篇幅中记录下来的。美国的基础研究生课程充满了研究工作的副产品，我们试图理解所有科学家普遍使用的这些数学合成的组成部分。然而，该系列的潜在可总结性在很大程度上仍然是一个谜。这个项目寻求在两年前四名研究人员解决了多重傅里叶级数中最著名的悬而未决的问题之一的重大突破的基础上再接再厉。***

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. ***