Mathematical Sciences: Some Probabilistic Aspects of Harmonic Analysis

界面上的几个数学问题 调和分析和概率论之间的联系 在这个项目中解决。概率论的应用 数学分析的历史悠久， 成功地重新调整了观点， 强大的方法。 这项工作的目标之一是审查 缺项三角级数的欣钦不等式。 这 不等式限制了Rademacher展开式的p范数， 函数由系数的和（平方）组成。 相同类型 如果使用缺项展开，则不等式的解成立。 努力 将表明，在下限常数可以是 估计得足够精确，以测量其对p阶的依赖性 动力. 第二条调查路线涉及的尾巴法律的 调和函数的迭代对数。这一规律，被发现 Kolmogorov在1950年提出的，测量三角函数的部分和 级数与其系数的平方和的关系。 尾部 法律是类似的，除了它只比较尾部的 系数的序列和尾部。 最近类似的结果 已获得其中调和函数比较与 他们的梯度的迭代图。目前的工作旨在 给出相应尾部比较的估计。 其他工作试图将指数的发展 经典的Littlewood-Paley理论。

Several mathematical problems situated at the interface between harmonic analysis and probability theory will be addressed in this project. The application of probability theory to questions of mathematical analysis has a rich history of success in refocusing the point of view and providing new, powerful methods. Among the targets of this work is an examination of Khintchine's inequality for lacunary trigonometric series. This inequality bounds the p-norm of the Rademacher expansion of a function by sums of (squares of) the coefficient. The same type of inequality obtains if a lacunary expansion is used. Efforts will be made to show that the constant in the lower bound can be estimated accurately enough to measure its dependence on the p-th power. A second line of investigation concerns tail laws of the iterated logarithm for harmonic functions. This law, discovered by Kolmogorov in 1950, measures partial sums of trigonometric series against the sum of squares of its coefficients. The tail law is similar except that it compares only the tail of the series and tail of the coefficients. Recently analogous results have been obtained in which harmonic functions are compared with iterated logarithms of their gradients. Current work seeks to give estimates of corresponding tail comparison. Other work seeks to relate developments of exponential martingales with classical Littlewood-Paley theory.