Approximation of maps and related results in the real setting

Wewill present (in two sessions) a survey concerning approximation of continuous and differentiable maps f : X → Y (between subsets of real affine spaces) mainly by differentiable and analytic maps. The starting point will be classical results of Stone–Weierstrass (polynomial approximation) and Whitney (analytic approximation) and some classical Extension related problems (when Y is itself an affine space). The central point of the survey will be the development of techniques to keep the prescribed target space Y during the approximation process. A mail tool to approach the Cr differentiable case (also in the situation when r is infinite) is the use of Cr triangulations of subsets Y that have either a polyhedral structure or more generally a subanalytic structure. Such Cr triangulations when Y is a Cr manifold go back to classical results of Cairns and Whitehead. When dealing with approximation by analytic maps, we will present some results concerning the case when Y is an analytic manifold with corners and we will explain what could be the crucial point to obtain further results.