Mathematical Sciences: Oscillatory Integrals and ConvolutionOperators

抽象的奥伯林 这是一个傅立叶分析的项目。它关注的问题与某些运营商和某些振荡积分自然与这些运营商。这些算子是通过与欧氏空间中曲线上的测度进行卷积而得到的。在最简单的情况下，振荡积分是一维积分， 具有多项式相函数的指数被积函数。这些积分自然地作为定义卷积算子的测度的傅里叶变换而出现。自1985年以来，人们在维度2和维度3中已经相当好地理解了这里感兴趣的问题。研究人员最近在4个维度上对这些问题取得了一些成功。本项目的目标是通过扩大第4方面所采用方法的范围和广度来继续这项工作。 纯数学传统上分为分析、代数和拓扑学。这是一个分析中的项目。粗略地说，分析的根源可以在微积分中找到。(The代数的根在高中代数中找到，拓扑的根在几何中找到。）微积分的研究对象是函数、导数和积分。 函数的导数是一个非常重要的工具-当它存在时。但并非所有函数都有导数。函数导数的存在性与函数的光滑性有关。平滑算子是一种将函数变换为密切相关但更平滑的函数的装置。（数学应用于 真实的世界，例如，流体力学中的问题，如飞机设计，几乎总是假设所涉及的函数具有一定程度的光滑性。通常情况下，当实际函数不是那么光滑时，它必须首先通过一个平滑算子。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。大多数平滑算子都是卷积算子。这个项目的动机是希望更好地理解这些卷积运算符。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.