ON ADAPTED COORDINATE SYSTEMS
ON ADAPTED COORDINATE SYSTEMS
复制标题
关于适应坐标系
DOI：
10.1090/s0002-9947-2011-04951-2
复制
发表时间：
2007
影响因子：
1.3
通讯作者：
D. Muller
中科院分区：
文献类型：
--
作者：
Isroil A.Ikromov;D. Muller
本文介绍了由V.I. Arnol'd，起着重要的作用，例如，在研究渐近展开的振荡积分。在二维空间中，A. N.瓦尔琴科给出了一个给定的坐标系的适应性的充分条件，并证明了存在一个适应的坐标系的解析函数没有多个组件。瓦尔琴科的证明是基于奇点的二维解析结果。在这篇文章中，我们提出了一个更基本的方法，这是基于给定的函数的根的Puixix级数展开。这种方法受到D. H. Phong和E. M. Stein关于牛顿多面体和振荡积分算子。它适用于任意实解析函数，甚至适用于任意有限型光滑函数。特别是，我们表明，Varchenko的条件实际上是必要的和充分的一个给定的坐标系的适应性和适应坐标总是存在于两个维度，即使在光滑，有限型设置。对于解析函数，D. H. Phong，E. M. Stein和J.A. Sturm关于实解析函数的增长性和稳定性，这是我们完成这篇论文后学到的。然而，与他们的工作相反，我们的证明更紧密地遵循Varchenko的算法，用于构建适应的坐标系，这对于平滑设置的扩展是有用的。
The notion of an adapted coordinate system for a given real-analytic function, introduced by V. I. Arnol'd, plays an important role, for instance, in the study of asymptotic expansions of oscillatory integrals. In two dimensions, A. N. Varchenko gave sufficient conditions for the adaptness of a given coordinate system and proved the existence of an adapted coordinate system for analytic functions without multiple components. Varchenko's proof is based on a two-dimensional resolution of singularities result. In this article, we present a more elementary approach to these results, which is based on the Puiseux series expansion of roots of the given function. This approach is inspired by the work of D. H. Phong and E. M. Stein on the Newton polyhedron and oscillatory integral operators. It applies to arbitrary real-analytic functions, and even to arbitrary smooth functions of finite type. In particular, we show that Varchenko's conditions are in fact necessary and sufficient for the adaptedness of a given coordinate system and that adapted coordinates always exist in two dimensions, even in the smooth, finite type setting. For analytic functions, a construction of adapted coordinates by means of Puiseux series expansions of roots has already been carried out in work by D. H. Phong, E. M. Stein and J. A. Sturm on the growth and stability of real-analytic function, as we learned after the completion of this paper. In contrast to their work, however, our proof more closely follows Varchenko's algorithm for the construction of an adapted coordinate system, which turns out to be useful for the extension to the smooth setting.