Differentiable approximation of continuous semialgebraic maps

In this work we approach the problem of approximating uniformly continuous semialgebraic maps f : S → T from a compact semialgebraic set S to an arbitrary semialgebraic set T by semialgebraic maps g : S → T that are differentiable of class C^ν for a fixed integer ν ≥ 1. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For ν = 1 we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For ν ≥ 2 we obtain density results in the following two relevant situations: either T is compact and locally C^ν semialgebraically equivalent to a polyhedron, for instance when T is a compact polyhedron; or T is an open semialgebraic subset of a Nash set, for instance when T is a Nash set. Our density results are based on a recent C^1-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.