Mathematical Sciences: Geometric Harmonic Analysis and Spectral Theory

这项工作的基础是存在于几何分析和谐波分析之间的紧密联系。后一个主题涉及将数学函数分解成更简单的部分，并从这些部分重构（合成）函数。通常，这些块自然地作为微分算子的特征函数出现，例如定义在曲面或流形上的拉普拉斯算子。然而，算子的本征函数出现在谐波分析思想刚刚开始渗透的许多情况下。本研究的目的是促进这一方向的进一步发展。一个阶数为3或更高的单连通李幂零群具有一个可表示为生成子平方和的次椭圆拉普拉斯算子。要确定这个算子的特征函数需要做很多工作。对于步骤2的群，如Heisenberg群，特征函数用Hermite函数来描述。目前的工作应该导致新的特殊函数类。其他工作将研究分数Hausdorff维数集。这些集长期以来在谐波分析中起着重要的作用，现在在数学的许多领域中变得突出起来。这个项目的一个目标是更仔细地研究最近发现的傅里叶变换的均方积分和原始函数对一维测量的相同积分之间的不等式。特别地，工作将集中于通过考虑函数的傅里叶变换乘以该测度的渐近行为来确定这种测度的支持的分形性质。一条新的研究路线将采用贝西科维奇在本世纪初提出的几乎周期函数的经典理论。该理论的缺点之一是，当为这些函数的空间定义范数时，它会产生一个很大的零空间。这项工作将寻找在空间上产生度量的新方法，发展一种分布观点，并推导出空间上广义谐波分析的有用理论，特别强调傅里叶变换的展开。

The basis for this work is the strong connections which exist between geometric and harmonic analysis. The latter subject concerns the decomposition of mathematical functions into simpler pieces and the reconstruction (synthesis) of functions from these pieces. Often the pieces arise naturally as eigenfunctions of differential operators such as the Laplace operator defined on a surface or manifold. However, eigenfunctions of operators occur in many contexts where the ideas of harmonic analysis are just beginning to penetrate. The object of this research is to foster further development in this direction. A simply-connected Lie nilpotent group of step 3 or higher has a sub-elliptic Laplacian representable as the sum of the squares of the generators. Work will be done in determining the eigenfunctions of this operator. For step 2 groups such as the Heisenberg group, the eigenfunctions are described in terms of Hermite functions. The present work should lead to new classes of special functions. Other work will study sets of fractional Hausdorff dimension. These sets which have long played a role in harmonic analysis are now coming into prominence in many areas of mathematics. One goal of this project will be to examine more closely recently discovered inequalities between the mean-square integral of Fourier transforms and the same integral of the originating function taken with respect to alpha-dimensional measure. In particular, work will concentrate on determining what fractal properties of the support of such a measure can be obtained by considering the asymptotic behavior of Fourier transforms of functions multiplied against the measure. A new line of investigation will take up the now classical theory of almost periodic functions developed early in the century by Besicovitch. One of the drawbacks to the theory was the fact that when a norm was defined for the space of such functions, it produced a large null space. This work will look for new ways of producing metrics on the space, developing a distributional point of view and deriving a useful theory of generalized harmonic analysis on the space, with special emphasis placed on the expansion of the Fourier transform.