Uniqueness for Multiple Trigonometric Series

Ash-Wang，Ash和Wang的第一个主要目标是推广Victor Shapiro和Jean Bourain关于球收敛的多重三角级数表示的唯一性的定理。Shapiro证明了，如果一个重三角级数处处Abel球面可和为一个可积函数，并且如果Shapiro的条件成立，那么它就是该函数的傅里叶级数。夏皮罗的条件表明，当半径趋于无穷大时，位于单位厚度环空中的系数的绝对值之和与半径之和的比率趋于零。更自然的条件是康奈斯的条件：当半径趋于无穷大时，球面上系数的平方和趋于零。由于Connes条件是处处收敛的结果，当Bourain证明了球面收敛到零的多重三角级数是零函数时，他能够避免任何系数增长假设。阿什和王将试图证明夏皮罗的结果，在假设中，康尼斯的病情取代了夏皮罗的病情。该定理的一个推论是三角级数的球面唯一性，该三角级数处处收敛于一个可积函数。此外，Ash和Wang希望通过允许一个不假设收敛的特殊集合来减轻到处收敛的假设。这样的集合称为唯一性集合。Ash和Wang想要证明所有可数集和某些不可数集都是唯一性集。通过一种称为多重傅立叶分析的过程，几乎任何曲面都是由更简单的曲面组成的。在纯数学中，一个长期存在的主要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。这个问题大约有六种主要的变种，这取决于如何将较简单的曲面组合成一般曲面。特别地，Ash和Wang将尝试确定平方收敛的二重三角级数是否具有唯一性。Ash和Wang还试图证明，在考虑球收敛的多重三角级数的唯一性问题时，某些薄集可以被忽略。既然我们生活在一个由空间和时间组成的四维世界中，那么研究高维版本的表面也是很有必要的。这样的物体称为流形。正如曲面与双三角级数相关联一样，流形也与多个三角级数相关联。因此，Ash和Wang还将尝试确定球收敛的多重三角级数是否具有唯一性。

ABSTRACT Ash-Wang Ash and Wang first major goal is to generalized theorems of Victor Shapiro and Jean Bourgain concerning uniqueness of representation by spherically convergent multiple trigonometric series. Shapiro proved that if a multiple trigonometric series is everywhere Abel spherically summable to an integrable function and if Shapiro's condition holds, then it is the Fourier series of that function. Shapiro's condition states that the ratio of the sum of the absolute values of the coefficients lying in an annulus of unit thickness to the radius tends to zero as the radius tends to infinity. A more natural condition is Connes' condition: that the sum of the squares of the coefficients lying on the surface of a sphere tends to zero as the radius tends to infinity. Since Connes' condition is a consequence of everywhere convergence, Bourgain was able to avoid any coefficient growth assumptions when he proved that a multiple trigonometric series everywhere spherically convergent to zero is the zero function. Ash and Wang will try to prove Shapiro's result with Connes' condition replacing Shapiro's condition in the hypothesis. A corollary of this theorem would be spherical uniqueness for trigonometric series that converge everywhere to an integrable function. Furthermore, Ash and Wang would like to lighten the hypothesis of everywhere convergence, by allowing an exceptional set on which convergence is not assumed. Such a set is called a set of uniqueness. Ash and Wang would like to show that all countable sets and certain uncountable sets are sets of uniqueness. Almost any surface is composed of simpler ones by a process called multiple Fourier analysis. A major long standing problem in pure mathematics is to show that this construction can be accomplished in only one way. This is called the problem of uniqueness. There are about a half dozen main varieties of this problem depending on just how the simpler surfaces are c ombined to make the general surface. In particular, Ash and Wang will try to determine if uniqueness holds for square convergent double trigonometric series. Ash and Wang will also try to show that certain thin sets may be ignored when considering the question of uniqueness for spherically convergent multiple trigonometric series. Since we live in a four dimensional world of space and time, it is also necessary to study a higher dimensional version of a surface. Such an object is called a manifold. Just as surfaces are associated with double trigonometric series, manifolds are associated with multiple trigonometric series. Thus, Ash and Wang will also try to determine if uniqueness holds for spherically convergent multiple trigonometric series.