Mathematical Sciences: Investigations in Analysis

数学科学：分析研究

• 批准号: 8707044
• 负责人: Daniel Oberlin
• 金额: $ 3.65万
• 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: 1987
• 资助国家: 美国
• 起止时间: 1987-10-15 至 1989-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8707044&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Investigations; Analysis

The study of functions defined on real n-space (i.e. the situation in which a quantity depends jointly on some number n of variable quantities) has been one of the basic endeavors of mathematics for centuries, with frequent feed- back from applications. For example, suppose we have an inhomogeneous body situated in 3-space; density within the body may be considered as a function of position. In practice, e.g. in computed tomography, the density function is unknown and must be explored indirectly. The mathematical procedure for doing this is to "transform" the unknown function into another function that one can more easily see. (A CAT scanner is an expensive and elaborate implementation of a particular sort of transform.) Mathematical investigation of a given transform typically addresses questions such as the following. How much information is lost in forming the new function from the old? What kind (from very smooth to extremely chaotic) of new function does one obtain from what kind of old function? If you change the old function just a little, does the new, transformed function also change just a little, or drastically? (In mathematical language, is the transform continuous, or not?) Professor Oberlin's project has mostly to do with the question of continuity for a family of transforms indexed by real numbers between 1 and infinity. At the two extremes, one has, respectively, the Riesz potential and the spherical maximal operator, both of whose behavior is well-understood. The problem is, what happens in between? This is a hard question. One can get a feel for the situation by means of computer experiments, but these are necessarily inconclusive. Intricate and clever arguments by the mathematician himself are required to settle matters finally.

对定义在实n空间上的函数的研究（即一个量共同依赖于变量的某个数n的情况）是几个世纪以来数学的基本努力之一，经常得到应用的反馈。例如，假设我们在三维空间中有一个非齐次物体；体内的密度可以看作是位置的函数。在实际应用中，例如在计算机断层扫描中，密度函数是未知的，必须间接地探索。这样做的数学过程是将未知函数“转换”为另一个更容易看到的函数。（CAT扫描仪是一种昂贵而复杂的特定转换实现。）对给定变换的数学研究通常会解决以下问题。在由旧函数形成新函数的过程中丢失了多少信息？从什么样的旧函数得到什么样的新函数（从非常平滑到极其混乱）？如果你稍微改变旧的函数，新的，变换后的函数是否也会改变一点点，或者改变很大？（用数学语言来说，这个变换是连续的还是不连续的？）Oberlin教授的项目主要是关于一组变换的连续性问题，这些变换以1到无穷之间的实数为索引。在两个极端情况下，分别有Riesz势和球面极大算子，它们的行为都是很容易理解的。问题是，中间会发生什么？这是个很难回答的问题。人们可以通过计算机实验来了解情况，但这些实验必然是不确定的。要最终解决问题，需要数学家本人进行复杂而巧妙的论证。