Harmonic Analysis on Spaces of Homogeneous Type
Harmonic Analysis on Spaces of Homogeneous Type
复制标题
DOI：
10.1007/978-3-540-88745-4
复制
发表时间：
2008-12
期刊：
影响因子：
--
通讯作者：
D. Deng;Yongsheng Han;Y. Meyer
中科院分区：
文献类型：
--
作者：
D. Deng;Yongsheng Han;Y. Meyer
这本书的书名可以定为《分析与几何》。作者正在解决以下问题：是否有可能在一个集合上执行一些调和分析？群的调和分析有着悠久的传统。在这里，我们给出了一个度量集X与（正）波莱尔措施？并且我们想构造一些在经典设置中依赖于傅立叶变换的算法。不用说，傅里叶变换不存在于任意度量集上。这种奋进不是一场革命。这是一个世纪前开始的一系列研究的延续，有两篇基础论文，我想讨论一下布里？y.的？第一篇论文是阿尔弗雷德·哈尔的博士论文，于1907年7月提交给哥廷根大学。在那个时候，它是已知的傅里叶级数展开的连续函数可能会发散在一个给定的点。Haar想知道这种现象是否发生在L [0，1]的每2个正交基上。他回答了这个问题，建设一个正交基础（今天被称为哈尔基础）的财产，扩大（在此基础上）的任何连续函数一致收敛到该功能。
This book could have been entitled “Analysis and Geometry.” The authors are addressing the following issue: Is it possible to perform some harmonic analysis on a set? Harmonic analysis on groups has a long tradition. Here we are given a metric set X with a (positive) Borel measure? and we would like to construct some algorithms which in the classical setting rely on the Fourier transformation. Needless to say, the Fourier transformation does not exist on an arbitrary metric set. This endeavor is not a revolution. It is a continuation of a line of research whichwasinitiated, acenturyago, withtwofundamentalpapersthatIwould like to discuss brie? y. The? rst paper is the doctoral dissertation of Alfred Haar, which was submitted at to University of Gottingen ̈ in July 1907. At that time it was known that the Fourier series expansion of a continuous function may diverge at a given point. Haar wanted to know if this phenomenon happens for every 2 orthonormal basis of L [0, 1]. He answered this question by constructing an orthonormal basis (today known as the Haar basis) with the property that the expansion (in this basis) of any continuous function uniformly converges to that function.