Mathematical Sciences: Oscillatory Integrals and ConvolutionOperators

数学科学：振荡积分和卷积算子

• 批准号: 9530537
• 负责人: Daniel Oberlin
• 金额: $ 4.88万
• 依托单位: Florida State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 1999-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9530537&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Oscillatory; Integrals; ConvolutionOperators

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.

摘要 Oberlin 这是傅立叶分析中的一个项目。它涉及与某些算子以及与这些算子自然相关的某些振荡积分相关的问题。算子是通过与欧几里德空间中曲线上的度量进行卷积给出的。在最简单的情况下，振荡积分是具有多项式相位函数的指数被积函数的一维积分。这些积分随着定义卷积算子的度量的傅立叶变换而自然产生。自大约 1985 年以来，这里感兴趣的问题在 2 维和 3 维中已经得到了相当好的理解。研究者最近在 4 维中的这些问题上取得了一些成功。该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、代数和拓扑领域。这是一个正在分析的项目。粗略地说，分析的根源可以在微积分中找到。 （代数的根源在于高中代数，拓扑的根源在于几何。）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——如果它存在的话。但并非所有函数都有导数。函数导数的存在与该函数的平滑度的概念密切相关。平滑算子是一种将函数转换为紧密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。而通常情况下，实际函数并不是这样的要平滑，它必须首先通过平滑算子。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水泥的过程相关。）大多数平滑算子属于卷积算子类型。该项目的动机是希望更好地理解这些卷积算子中的某些。标题中的振荡积分只是帮助理解这一点的工具。