Integration of Oscillatory and Subanalytic Functions

TL;DR

This paper proves the stability of a class of functions, including subanalytic and oscillatory functions, under integration and Fourier transform, extending previous work and highlighting interactions between analysis and singularity theory.

Contribution

It extends the stability results of subanalytic functions to include oscillatory functions, enriching the framework for analysis and singularity theory.

Findings

Stability under integration of oscillatory and subanalytic functions

Stability under Fourier transform of these functions

Extension of previous theoretical frameworks

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.