Approximation theorems for Nash mappings and Nash manifolds

Abstract:

Let $0 \leq r < \infty$. A Nash function on ${\mathbf {R}^n}$ is a ${C^r}$ function whose graph is semialgebraic. It is shown that a ${C^r}$ Nash function is approximated by a ${C^\omega }$ Nash one in a strong topology defined in the same way as the usual topology on the space $\mathcal {S}$ of rapidly decreasing ${C^\infty }$ functions. A ${C^r}$ Nash manifold in ${\mathbf {R}^n}$ is a semialgebraic ${C^r}$ manifold. We also prove that a ${C^r}$ Nash manifold for $r \ge 1$ is approximated by a ${C^\omega }$ Nash manifold, from which we can classify all ${C^r}$ Nash manifolds by ${C^r}$ Nash diffeomorphisms.