On adapted coordinate systems

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 arbi...

Full description

Saved in:
Bibliographic Details
Published in:Transactions of the American Mathematical Society 2011-06, Vol.363 (6), p.2821-2848
Main Authors: IKROMOV, ISROIL A., MÜLLER, DETLEF
Format: Article
Language:English
Subjects:
Analytic functions
Coordinate systems
Exact sciences and technology
Functions of a complex variable
General mathematics
General, history and biography
Integral equations
Mathematical analysis
Mathematical functions
Mathematical integrals
Mathematics
Polyhedrons
Polynomials
Real functions
Roots of functions
Sciences and techniques of general use
Series expansion
Taylor polynomials
Vertices
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-2011-04951-2