On smooth locally o-minimal functions (Q491440)

Let \(\mathcal M\) be an o-minimal expansion of the real field. The author calls a subset \(X \subseteq \mathbb R^n\) an \(\mathcal S\)-set, if for every \(x \in \mathbb R^n\) there is an open ball \(B\) with center \(x\) such that \(X \cap B\) is definable in \(\mathcal M\) with parameters in \(\mathbb R\). A continuous function is an \(\mathcal S\)-function if its graph is an \(\mathcal S\)-set. The collection of \(\mathcal S\)-sets satisfies the axioms of Shiota's theory of \(\mathfrak X\)-sets [\textit{M. Shiota}, Geometry of subanalytic and semialgebraic sets. Boston, MA: Birkhäuser (1997; Zbl 0889.32006)]. The paper under review is dedicated to the study of \(C^\infty\)-smooth \(\mathcal S\)-functions and \(\mathcal S\)-manifolds for \(\mathcal M\) being an o-minimal expansion of the real exponential field admitting \(C^\infty\) cell decomposition. Since in general sums, products, and composites of \(\mathcal S\)-functions are not \(\mathcal S\)-functions, certain better behaved subclasses of \(\mathcal S\) are introduced: the class \(\mathcal S_u\) of \(\mathcal S\)-functions which map bounded sets to bounded sets, and the class \(\mathcal S_{\mathrm{sld}}\) of the localization of \(\mathcal S_u\) by positive \(\mathcal S_u\)-functions. The author obtains results on \(C^\infty\) approximation of differentiable \(\mathcal S_{\mathrm{sld}}\)-functions, partitions of unity, and separation of closed \(\mathcal S\)-sets. He studies \(\mathcal S_{\mathrm{sld}}\)-diffeomorphisms and their smoothing as well as different diffeomorphism classes of \(\mathcal S\)-manifolds. The final section is devoted to sums and composites of \(\mathcal S\)-functions.