2-microlocal Formalism
2-microlocal Formalism
复制标题
2-微局部形式主义
DOI：
10.1090/pspum/072.2/2112123
复制
发表时间：
2003
影响因子：
0.8
通讯作者：
S. Seuret
中科院分区：
文献类型：
--
作者：
J. L. Véhel;S. Seuret
本文致力于研究一种度量分布局部正则性的精细方法。从J.M.Bony引入的2-微局域分析出发，我们发展了一个2-微局域的形式主义，这在很大程度上符合多重分形主义的精神。这允许定义一个新的正则性函数，我们称之为2-微局域谱。2微局域谱被证明是一个强大的工具，我们在三个方向上应用。首先，它允许恢复所有已知的关于局部正则性指数的结果，以及发现关于它们的新性质。其次，2-微局地光谱提供了对2-微局地边界的更深层次的理解。它产生了一种特别自然的方式，在可数不清的密集点集上规定这些边界。最后，我们探讨了多重分形论和2-微局域论之间的相似之处。文中以魏尔斯特拉斯函数、黎曼函数以及空位小波级数为例说明了这些应用。
This paper is devoted to the study of a fine way to measure the local regularity of distributions. Starting from the 2-microlocal analysis introduce- d by J.M. Bony, we develop a 2-microlocal formalism, much in the spirit of the multifractal formalism. This allows to define a new regularity function, that we call the 2-microlocal spectrum. The 2-microlocal spectrum proves to be a powerful tool that we apply in three directions. First, it allows to recover all previously known results on local regularity exponents, as well as to discover new properties about them. Second, the 2-microlocal spectrum provides a deeper understanding of the 2-microlocal frontiers. It yields in particular a natural way of prescribing these frontiers on a countable dense set of points. Finally, we explore the close parallel between the multifractal and 2-microlocal formalisms. These applications are illustrated on examples such as the Weierstrass and the Riemann functions, as well as lacunary wavelet series.