Mathematical Sciences: Harmonic Analysis on Spaces of Homogeneous Type

从历史上看，经典谐波分析起源于对声和热传输等物理现象的建模问题，特别是基本传输模式的叠加。在数学上，这转化为更一般的概念，用简单的构造块来表示函数，在现代理论中，人们也研究广义函数和算子。经典地，构建块通常与底层空间的代数结构相关；然而，在当前的“健壮”理论中，它们可能是根据一类算子来选择的，并且底层域的结构可能不太好。大多数研究都是通过将一个空间带入另一个空间的算子来理解的。基础领域包括经典欧几里得空间和推广，如时空、李变换群、局部场和离散结构。其应用遍及整个数学领域，并广泛应用于天线设计、CAT扫描、断层扫描、地震学、粒子物理学、经济预测、模式识别和编码理论等领域。看似抽象的领域，如局部域和“树”，实际上是模式识别和数值近似应用的自然设置。Weiss教授和Taibleson教授在以下领域进行研究：harry空间理论，根据该理论，构建模块是原子和分子；利用“块”生成的空间收敛傅里叶级数；插值理论，它关注的是函数族的映射；以及树木谐波分析的发展。Weiss教授是谐波分析研究的领导者，该研究处理非代数结构的构建块，而Taibleson教授则是将经典谐波分析技术扩展到局部领域和树木的领导者。他们提出的研究包括将用于研究块的方法扩展到尚未很好理解的相关功能空间和属性；插值方法的扩展；将树的谐波分析扩展到更一般的结构，如图。后者将把这种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.