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Approximation theorems for Nash mappings and Nash manifolds. (English) Zbl 0601.58005
A semialgebraic submanifold M of $${\mathbb{R}}^ n$$ of class $$C^ r$$ $$(r=0,...,\infty,\omega)$$ is called a $$C^ r$$-Nash manifold, and a $$C^ r$$-Nash map is a $$C^ r$$-map of a semialgebraic graph. Using partitions of unity with Nash functions and stratifications by Nash manifolds, several results of fundamental importance are derived: Every $$C^ r$$- Nash map between $$C^ r$$-Nash manifolds can be approximated by a $$C^{\omega}$$-Nash map in the $$C^ r$$-topology. (A limit $$f_ k\to 0$$ in this topology means the uniform convergence $$v_ 1...v_ sf_ k\to 0$$ for any $$C^ r$$-Nash vector fields $$v_ 1,...,v_ s$$ in the number $$s\leq r.)$$ Every $$C^ r$$-Nash manifold M in $${\mathbb{R}}^ n$$ $$(1\leq r<\infty)$$ can be approximated by $$C^{\omega}$$-Nash manifolds in $$C^ r$$-topology, and for any compact $$C^{\omega}$$-Nash submanifold $$N\subset M$$, the approximations can be chosen identical on N. As a result, $$C^ r$$-Nash diffeomorphism classes are identical with $$C^{\omega}$$-Nash diffeomorphism classes of $$C^ r$$-Nash manifolds. (For the whole class of abstract Nash manifolds, an analogous result is not true.) Moreover, a $$C^ 0$$-vector bundle over a $$C^ r$$-Nash manifold (0$$\leq r\leq \omega)$$ possesses a unique $$C^ r$$-Nash bundle structure.
Reviewer: J.Chrastina

##### MSC:
 58A07 Real-analytic and Nash manifolds 14Pxx Real algebraic and real-analytic geometry
##### Keywords:
Nash functions; Nash manifolds; Nash diffeomorphism
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##### References:
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