Quantifier elimination and rectilinearization theorem for generalized quasianalytic algebras

We consider for every n is an element of N an algebra A(n) of germs at 0 is an element of R-n of continuous real-valued functions, such that we can associate to every germ f. A(n) a (divergent) series T (f) with non-negative real exponents, which can be thought of as an asymptotic expansion of f. We require that the R-algebra homomorphism f bar right arrow T(f) be injective (quasianalyticity property). In this setting, we prove analogue results to Denef and van den Dries' quantifier elimination theorem and Hironaka's rectilinearization theorem for subanalytic sets.