Mathematics – Classical Analysis and ODEs
Scientific paper
2007-06-07
Mathematics
Classical Analysis and ODEs
31 pages, 6 figures
Scientific paper
The notion of an adapted coordinate system, introduced by V.I.Arnol'd, plays an important role 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 a class of analytic functions without multiple components. Varchenko's proof is based on Hironaka's theorem on the resolution of singularities. In this article, we present a new, elementary and concrete approach to these results, which is based on the Puiseux series expansion of roots of the given function. Our method applies to arbitrary real analytic functions, and even extends to arbitrary smooth functions of finite type. Moreover, by avoiding Hironaka's theorem, we can give necessary and sufficient conditions for the adaptedness of a given coordinate system in the smooth, finite type setting.
Ikromov Isroil A.
Müller Detlef
No associations
LandOfFree
On adapted coordinate systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On adapted coordinate systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On adapted coordinate systems will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-442224