Mathematical Sciences: Oscillatory Integrals and ConvolutionOperators

这是傅里叶分析中的一个课题。它关注的是与某些算子有关的问题以及与这些算子自然相关的某些振荡积分问题。这些算子是由在欧几里德空间中的曲线上与测度的卷积给出的。在最简单的情况下，振荡积分是具有多项式相函数的指数积分的一维积分。这些积分是定义卷积算子的测度的傅里叶变换。我们感兴趣的问题从1985年开始就已经在二维和三维中得到了很好的理解。这位研究者最近在四维空间的这些问题上取得了一些成功。这个项目的目标是通过扩展维度4中使用的方法的范围和范围来继续这项工作。纯数学传统上分为分析、代数和拓扑学三个领域。这是一个分析项目。粗略地说，分析的根源可以在微积分中找到。（代数的根在高中代数中，拓扑学的根在几何中。）微积分的研究对象是函数、导数和积分。函数的导数是一个极其重要的工具——当它存在的时候。但不是所有的函数都有导数。一个函数导数的存在性与该函数的平滑性密切相关。平滑算子是一种将一个函数转换成一个密切相关但更平滑的函数的装置。(数学在现实世界中的应用，例如飞机设计等流体力学问题，几乎总是默认所涉及的函数具有一定程度的平滑性。通常情况下，当实际函数不是那么平滑时，必须首先通过平滑运算符传递它。平滑算子在通信理论中也非常有用，它们与噪声去除和图像增强过程有关。大多数平滑算子属于一种称为卷积算子的类型。这个项目的动机是希望更好地理解这些卷积算子。标题中的振荡积分只是帮助理解的工具。

Abstract Oberlin This is a project in Fourier analysis. It is concerned with problems related to certain operators and to certain oscillatory integrals which are naturally associated with those operators. The operators are given by convolution with measures on curves in Euclidean spaces. In the simplest case the oscillatory integrals are one-dimensional integrals with an exponential integrand having polynomial phase function. These integrals arise naturally as the Fourier transforms of the measures defining the convolution operators. The questions of interest here have been fairly well understood in dimensions 2 and 3 since about 1985. The investigator has recently had some success with these problems in 4 dimensions. The goal of this project is to continue that work by extending the range and scope of the methods employed in dimension 4. Pure mathematics is traditionally divided into the areas of analysis, algebra, and topology. This is a project in analysis. Very roughly, the roots of analysis are to be found in calculus. (The roots of algebra are found in high school algebra, and those of topology are in geometry.) The objects of study in calculus are functions, derivatives, and integrals. The derivative of a function is an extremely important tool- when it exists. But not all functions have derivatives. The existence of a function's derivative is tied up with the idea of that function's smoothness. A smoothing operator is a device which transforms a function into a closely related but smoother function. (Applications of mathematics to the real world, e.g., problems in fluid mechanics like airplane design, almost always make the tacit assumption that the functions involved possess a certain degree of smoothness. When, as is often the case, the actual function is not that smooth, it must first be passed through a smoothing operator. Smoothing operators are also extremely useful in communications theory, where they are associated with the processes of noise removal and image enhan cement.) Most smoothing operators are of a type known as convolution operators. The motivation for this project is the desire to understand better certain of these convolution operators. The oscillatory integrals of the title are just tools which aid in this understanding.