Mathematical Sciences: Investigations in Analysis

定义在真实的n-空间上的函数的研究（即 一个量共同取决于一些 变量的数量n）一直是基本的 几个世纪以来，数学的努力，频繁的饲料- 从应用程序中返回。例如，假设我们有一个 不均匀的身体位于3空间;密度内 身体可以被认为是位置的函数。在 密度函数的应用，例如在计算机断层摄影中 是未知的，必须间接探索。数学 这样做的程序是“转换”未知 功能转换成另一个更容易看到的功能。 （CAT扫描仪是一种昂贵而复杂的实现方式。 一种特殊的变换）。数学研究 一个给定的转换通常解决的问题，如 下面的.有多少信息在形成新的 功能从旧？什么样的（从非常光滑到非常光滑 一个新的功能是从什么样的 旧功能？如果你稍微改变一下旧的函数， 这个新的，转换后的函数是否也有一点点变化， 还是彻底地(In数学语言，是转换 连续，或不连续？） 奥伯林教授的项目主要与 一个变换族的连续性问题， 1到无穷大之间的真实的数。在两个极端， 分别有Riesz势和球势 最大算子，其行为都很好理解。 问题是，这中间会发生什么？这是一个很难 问题一个人可以通过 计算机实验，但这些必然是不确定的。 数学家自己提出的复杂而巧妙的论点 最终解决问题

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.