Uniqueness for Multiple Trigonometric Series

多重三角级数的唯一性

• 批准号: 9707011
• 负责人: Marshall Ash
• 金额: $ 9万
• 依托单位: DePaul University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9707011&HistoricalAwards=false
• 关键词: Uniqueness; Multiple; Trigonometric; Series

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.

摘要 阿什-王 Ash和Wang的第一个主要目标是推广维克托的定理 Shapiro和Jean Bourgain关于表示的唯一性 球面收敛的多重三角级数 夏皮罗 证明了如果一个多重三角级数处处Abel 球面可和的可积函数，如果Shapiro 条件成立，则它是该函数的傅里叶级数。 夏皮罗条件指出，绝对值之和的比率 位于单位厚度环中的系数值 当半径趋于无穷大时，半径趋于零。一个更自然的条件是康纳斯的条件： 位于球体表面的系数的平方趋于 当半径趋于无穷大时，自从康纳斯的情况 是处处收敛的结果，布尔甘能够 避免任何系数增长假设时，他证明， 多重三角级数处处球收敛于 zero是zero函数。阿什和王会试图证明夏皮罗的 结果与Connes的条件取代Shapiro的条件， 假说.该定理的推论是球面唯一性 对于处处收敛于可积的三角级数， 功能此外，阿什和王想减轻 假设处处收敛，通过允许一个例外集 在此不假设收敛。 这样的一个集合叫做 唯一性Ash和Wang想证明所有可数集 某些不可数集是唯一性集 几乎所有的表面都是由简单的表面通过一种称为多重傅立叶分析的过程组成的。一个主要的长期存在的问题，在纯 数学的目的是证明这个构造可以 只有一种方式。这就是所谓的唯一性问题。有 这个问题的六种主要类型取决于 简单曲面如何组合成一般曲面。 特别是，阿什和王将试图确定， 平方收敛双重三角级数成立。 阿什和王还将试图表明，某些薄集可能是 在考虑球面唯一性问题时忽略了 收敛多重三角级数 既然我们住在四楼 空间和时间的三维世界，也有必要研究 一个更高维度的曲面 这样的物体被称为 歧管。 正如曲面与双曲面 三角级数，流形与多重三角级数相关联。因此，阿什和王也将尝试 判断球面收敛的唯一性是否成立 多重三角级数