Mathematical Sciences: RUI Uniqueness of Multiple Trigonometric Series and Symmetric Analysis

该项目继续对涉及多个三角级数的数学问题进行合作研究。这项工作的起点在于1870年G.Cantor的结果，他证明至多一个三角级数可以收敛到给定的函数。这是一个不同于确定给定级数是否收敛的问题。它属于所谓的唯一性问题的范畴。多重三角级数的唯一性问题是一个长期存在的问题。这个问题有更多的障碍，因为现有的多种求和方法没有出现在一个变量的情况下。调查人员已经确定，在矩形求和的情况下，唯一性是有效的。现在将继续研究新开发的技术，以研究其他求和方法，如迭代或球面收敛。此外，还将进一步研究多重唯一性问题和确定三角级数(多变量)何时是傅立叶级数的问题。三角级数构成了现代调和分析理论的支柱。它们在从纯数论到物理偏微分方程解等分析问题的研究中扮演着无处不在的角色。关于独特性问题的最新结果开辟了一条研究可能性的新途径，这些可能性本质上已经休眠了几十年。

This project continues collaborative research on mathematical questions concerning multiple trigonometric series. The starting point of the work rests with the 1870 result of G. Cantor who showed that at most one trigonometric series can converge to a given function. This is a different question from that of determining whether a given series converges. It falls into the category of what is known as a uniqueness question. A longstanding problem has been the uniqueness question for multiple trigonometric series. There are many more obstacles to this question arising from the multiple methods of summation available, which do not show up in the one variable case. The investigators have determined that the uniqueness is valid in the case of rectangular summation. Work will now be done in carrying over newly developed techniques to investigate other summation methods such as iterated or spherical convergence. In addition, further studies will be carried out on questions of multiple uniqueness and the problem of determining when a trigonometric series (in several variables) is a Fourier series. Trigonometric series form the backbone of the modern theory of harmonic analysis. They play a ubiquitous role in the study of analytic problems ranging from pure number theory to the solution of partial differential equations of physics. Recent results on the uniqueness question has opened up a new avenue of research possibilities which have essentially been dormant for decades.