ON ADAPTED COORDINATE SYSTEMS

By Isroil A.Ikromov, Detlef Muller. 2007

58 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/80a818f5c1073240e431bb633bd6e7f8acc39a85

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Uniform Estimates for the Fourier Transform of Surface Carried Measures in ℝ3 and an Application to Fourier Restriction (2010, background): “systems x = (x1,x2) at the origin, and where dx is the distance between the Newton polyhedron and the origin in the coordinates x. A given coordinate system xis said to be adapted to φif h(φ) = dx.In [11] we proved that one can always ﬁnd an adapted local coordinate system in two dimensions, thus ge”
Oscillatory integral decay, sublevel set growth, and the Newton polyhedron (2008, methodology): “These results were generalized to smooth phase in [ 8 ].”
A multi-dimensional resolution of singularities with applications to analysis (2010, methodology): “Acknowledgements The third author thanks Professor Detlef Müller for valuable comments and Professor Kalle Karu for mentioning a reference for Lemma 2.4.”
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (2016, background): “• Fourier restriction for hyperbolic hypersurfaces • Fourier restriction for manifolds of higher codimensions • Fourier restriction for measures supported on certain fractal sets (such as Cantor type sets) satisfying uniform Fourier decay estimates • Uniform Fourier restriction estimates with respec”
Estimates for maximal functions associated with hypersurfaces in ℝ3 and related problems of harmonic analysis (2010, background): “In the case of an analytic function φ, the function ψ can be chosen to be analytic, in which case it can be identified with one of the roots of φ, called the principal root, or a partial sum of the Puiseux series expansion of it (see [17], and compare also Section 3), which is why we called it the p”