Mathematical Sciences: Harmonic Analysis on Spaces of Homogeneous Type

在历史上，经典的谐和分析起源于对诸如声和热的传输这样的物理现象进行建模的问题，特别是传输的基本模式的叠加。在数学上，这转化为用简单的积木表示函数的更一般的概念，在现代理论中，人们也研究广义函数和运算符。传统上，构建块通常与底层空间的代数结构相关；然而，在当前的“稳健”理论中，它们可能是相对于一类算子而选择的，并且底层域可能结构不太好。从运算符将一个空间转移到另一个空间的角度来看，大多数研究是可以理解的。潜在的领域包括经典的欧几里德空间和推广，如时空，李变换群，局域场，和离散结构。应用程序在整个数学中无处不在，在天线设计、CAT扫描、层析成像、地震学、粒子物理、经济预测、模式识别和编码理论等广泛的应用程序中也是如此。看似抽象的领域，如局部领域和“树”，实际上是模式识别和数值逼近应用的自然环境。魏斯和泰布尔森教授从事以下领域的研究：Hary空间理论，其构建块是原子和分子；傅立叶级数通过“块”生成的空间的收敛；内插理论，它涉及函数族的映射；以及树上调和分析的发展。Weiss教授是研究调和分析的领导者，调和分析处理的是不是从代数结构派生的积木，而泰布尔森教授是将经典调和分析技术扩展到局部场和树的领导者。他们提出的研究包括将块研究中使用的方法扩展到尚未很好理解的相关函数空间和性质；内插方法的扩展；以及将树上的调和分析扩展到更一般的结构，如图。后者将把这一方法应用于概率论和组合学。

Historically, classical harmonic analysis originates with the problem of modeling such physical phenomena as the transmission of sound and heat, and in particular the superposition of fundamental modes of transmission. Mathematicaly, this translates into the more general idea of the representation of functions in terms of simple building blocks, and in the modern theory one studies generalized functions and operators as well. Classically, the building blocks would typically be related to the algebraic structure of the underlying space; however, in current "robust" theory they may be chosen in relation to a class of operators, and the underlying domain may be less well structured. Most research is understandable in terms of operators that carry one space into another. The underlying domains include classical Euclidean spaces and generalizations such as space-time, Lie transformation groups, local fields, and discrete structures. Applications are pervasive throughout mathematics, and in applications as wide ranging as antenna design, CAT scans, tomography, seismology, particle physics, economic forecasting, pattern recognition, and coding theory. Seemingly abstract domains such as local fields and "trees" are, in fact, natural settings for applications in pattern recognition and numerical approximation. Professors Weiss and Taibleson conduct research in the following areas: Hary space theory, in terms of which the building blocks are atoms and molecules; convergence of Fourier series by means of spaces generated by "blocks"; interpolation theory, which is concerned with mappings of families of functions; and the development of harmonic analysis on trees. Professor Weiss is a leader in the study of harmonic analysis that treats building blocks that do not derive from algebraic structure, while Professor Taibleson is a leader in the extension of classical harmonic analysis techniques to local fields and trees. Their proposed research includes the extension of methods, used in the study of blocks, to related function spaces and properties that are not yet well understood; extension of interpolation methods; and the cexpansion of harmonic analysis on trees to more general structures such as graphs. The latter would bring this methodology to bear in the context of probability and combinatorics.