Integration of Oscillatory and Subanalytic Functions
Abstract
We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This paper extends the investigation started in [J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under integration of sums of products of real globally subanalytic functions and their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.
Cite
@article{arxiv.1601.01850,
title = {Integration of Oscillatory and Subanalytic Functions},
author = {Raf Cluckers and Georges Comte and Daniel J. Miller and Jean-Philippe Rolin and Tamara Servi},
journal= {arXiv preprint arXiv:1601.01850},
year = {2018}
}
Comments
Final version. Accepted for publication in Duke Math. Journal. Changes in proofs: from Section 6 to the end, we now use the theory of continuously uniformly distributed modulo 1 functions that provides a uniform technical point of view in the proofs of limit statements