Propriétés oscillatoires des intégrales de [formula omitted]-fonctions

Abstract

We deal here with a o-minimal class of real functions called x λ -functions. They are finite compositions of subanalytic maps and power maps. These functions have nice properties of non-oscillation: the number of connected components of a 1-parameter family of x λ -functions is uniformly bounded with respect to the parameter if the dependency is analytic (or more generally o-minimal) in the parameter. Our purpose is to study how this kind of property can remain under integration. Some result of this kind have been already proved for the smaller class of subanalytic functions. We show here that for almost all power maps arising in a x λ -function, the integration leads to a non-oscillating function. Nevertheless, this class of functions is not stable under integration.