Uniqueness for Multiple Trigonometric Series

Ash和Wang的第一个主要目标是推广Victor Shapiro和Jean Bourgain关于球收敛多重三角级数表示的唯一性定理。夏皮罗证明了如果一个多重三角级数处处都能被阿贝尔球求和为一个可积函数如果夏皮罗的条件成立，那么它就是那个函数的傅里叶级数。夏皮罗条件表明，当半径趋于无穷大时，单位厚度环空中各系数绝对值之和与半径之比趋于零。一个更自然的条件是cones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的条件是处处收敛的结果，当布尔甘证明一个处处球收敛于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。Ash和Wang将尝试用cones条件代替假说中的Shapiro条件来证明Shapiro的结果。这个定理的一个推论是，处处收敛于可积函数的三角级数的球唯一性。此外，Ash和Wang希望通过允许一个不假设收敛的例外集来减轻处处收敛的假设。这样的集合称为唯一性集合。Ash和Wang想证明所有可数集合和某些不可数集合都是唯一性集合。几乎所有的曲面都是由更简单的曲面通过多次傅立叶分析组成的。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体，这取决于如何将更简单的表面组合成一般的表面。特别是，Ash和Wang将尝试确定是否唯一性的平方收敛的二重三角级数。Ash和Wang还将试图证明在考虑球收敛多重三角级数的唯一性问题时，某些细集可以被忽略。由于我们生活在一个四维空间和时间的世界中，所以也有必要研究一个更高维度的表面。这样的对象称为流形。正如曲面与二重三角级数有关一样，流形与多重三角级数有关。因此，Ash和Wang也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.