Multiple Trigonometric Series and Multiple Walsh Series

建议摘要二重三角级数的平方部分和是指两个指数都小于或等于某一固定值的所有项的和。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序列处处收敛于零，则该级数必然是平凡级数。如果这是真的，那么我们将尝试将这一结果推广到更高的维度。2维的Shapiro和高维的Bourain，以及Ash-Freling-Rinnee和Tetunashvili对任意维的无限矩形收敛情形，分别给出了相应的圆/球收敛公式。有一些证据表明，平方唯一性实际上可能是错误的。例如，存在一个系数增长速度快于多项式的处处平方收敛的重三角级数。我们的第二个目标是研究一个相关问题：多个Walsh级数在不同类型求和模式下的唯一性。我们将采取的新方法是使用经典的调和分析方法。通过将传统的鞅方法与最近在多重三角级数领域取得的进展发展起来的新技术相结合，我们期待在这方面取得更大的进展。这反过来可能对三角级数的平方唯一性问题有更深入的了解。第三个目标是研究关于平方可积函数的傅里叶级数的逐点圆收敛这一长期悬而未决的问题。我们将试图通过研究双沃尔什级数的相应问题来阐明这一点，双沃尔什级数是树指标的特殊形式和两个参数鞅，几乎任何曲面都是由更简单的曲面组成的，这一过程称为多重傅立叶分析。纯数学中一个长期存在的重大问题是，证明这种构造只能以一种方式完成。这被称为唯一性问题。这个问题大约有六个主要的变种，这取决于简单的曲面是如何组合成一般曲面的。以不同的顺序聚集简单曲面可能会产生不同的结果曲面。对于某些采集过程，已经证明了唯一性。也就是说，对于这样的聚集，只有一种方法来产生合成表面。唯一性仍然是一个悬而未决的问题的最重要的收集过程称为平方收敛。我们将尝试确定此过程是否具有唯一性。另一种构建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，理解这两种构建方法中的一种可能会导致对另一种方法的深入了解。

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.