Multiple Trigonometric Series and Multiple Walsh Series

摘要：二重三角级数的平方和是所有两个指标都小于或等于固定值的项的和。我们的第一个目标是研究二重三角级数的平方唯一性。我们的意思是，如果一个二重三角级数的平方部分和的序列处处收敛于零，那么这个级数必然是平凡级数。如果这是真的，我们将尝试把这个结果推广到更高的维度。在2维的Shapiro和高维的Bourgain分别给出了圆/球收敛性的相应表述；以及由Ash-Freiling-Rinneand独立地，Tetunashvili提出的任意维度的无限制矩形收敛情况。有一些证据表明正方形唯一性实际上可能是错误的。例如，存在一个处处平方收敛的双三角级数，其系数的增长率大于多项式。我们的第二个目标是研究一个相关的问题：不同类型和模下多个Walshseries的唯一性。我们将采用的新方法是使用经典的谐波分析方法。通过将传统的鞅方法与最近在多重三角级数领域发展起来的新技术相结合，我们期望在这里取得很大的进展。这反过来又可以使我们对三角级数的平方唯一性问题有所了解。第三个目标是研究平方可积函数的傅里叶级数的点向圆收敛性这一长期悬而未决的问题。我们将通过研究树索引和双参数鞅的特殊形式的双Walsh级数的相应问题来阐明这一点。几乎任何表面都是由更简单的表面组成的，这一过程被称为多重傅立叶分析。纯数学中一个长期存在的主要问题是证明这种结构只能以一种方式完成。这就是所谓的唯一性问题。这个问题大约有六种主要的变体，这取决于如何将更简单的表面组合成一般的表面。按不同的顺序聚集简单曲面，可以得到不同的合成曲面。对于某些收集程序，唯一性已得到证明。也就是说，对于这样的聚集，只有一种方法来产生结果曲面。唯一性仍然是一个悬而未决的问题的最重要的收集过程被称为平方收敛。我们将尝试确定唯一性是否适用于此过程。另一种构造表面的方法是将其构建为二维振荡方波的组合。这样的过程称为多重沃尔什级数。我们也将在此背景下考虑唯一性问题。我们希望通过理解其中一种构建方法，可以对另一种构建方法有所启发。

Proposal AbstractA square partial sum of a double trigonometric series is the sum of all theterms with both indices less than or equal to a fixed value. Our first goal is tostudy square uniqueness for double trigonometric series. By this we mean that ifthe sequence of square partial sums of a double trigonometric series converges tozero everywhere, then the series is necessarily the trivial series. If this is true, wewill then try to generalize this result to higher dimensions. The correspondingstatements for circular/spherical convergence have been shown by Shapiro indimension 2 and by Bourgain in higher dimensions; and by Ash-Freiling-Rinneand, independently, Tetunashvili for the unrestrictedly rectangular convergencecase in any dimension. There is some evidence that square uniqueness mayactually be false. For example, there is an everywhere square convergent doubletrigonometric series with coefficients having faster than polynomial growth rate.Our second goal is to study a related question: uniqueness for multiple Walshseries under different types of summation modes. The new approach we willtake is to use classical harmonic analysis methods. By combining the traditionalmartingale approach with the new techniques developed from recent progressmade in the area of multiple trigonometric series, we expect much progresscan be made here. This in turn may give insights into the square uniquenessquestion for trigonometric series. The third goal is to study the long standingopen question about the pointwise circular convergence for Fourier series ofsquare integrable functions. We will try to shed some light on this by studyingthe corresponding question for double Walsh series, which are special form oftree-index and two parameter martingales.Almost any surface is composed of simpler ones by a process called multipleFourier analysis. A major long standing problem in pure mathematics is toshow that this construction can be accomplished in only one way. This is calledthe problem of uniqueness. There are about a half dozen main varieties of thisproblem depending on just how the simpler surfaces are combined to make thegeneral surface. Gathering the simple surfaces in different orders may lead todifferent resultant surfaces. For certain gathering procedures, uniqueness hasbeen proved. That is, for such gathering, there is only one way to producethe resultant surface. The most important gathering procedure for which theuniqueness remains an open question is called square convergence. We will tryto determine if uniqueness holds for this procedure. Another way to construct asurface is to build it up as a combination of two dimensional oscillating squarewaves. Such a process is called a multiple Walsh series. We will consider theuniqueness question in this context also. We hope that understanding one ofthe two methods of construction may lead to insights about the other.