Efroymson's Approximation Theorem asserts that if f is a ${\mathcal{C}}^0$ semialgebraic mapping on a ${\mathcal{C}}^{\infty }$ semialgebraic submanifold M of ${\mathbb{R}}^n$ and if $\epsilon :M\to \mathbb{R}$ is a positive continuous semialgebraic function then there is a ${\mathcal{C}}^{\infty }$ semialgebraic function $g:M\to \mathbb{R}$ such that $|f-g|<\epsilon$. We prove a generalization of this result to the globally subanalytic category. Our theorem actually holds in a larger framework since it applies to every function which is definable in a polynomially bounded o-minimal structure (expanding the real field) that admits ${\mathcal{C}}^{\infty }$ cell decomposition. We also establish approximation theorems for Lipschitz and ${\mathcal{C}}^1$ definable functions.

中文翻译：

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全局子分析和Denjoy-Carleman类的逼近

Efroymson的近似定理断言，如果f是a${\mathcal{C}}^0$ a上的半代数映射 ${\mathcal{C}}^{\infty }$semialgebraic子流形中号的${\mathbb{[R}}^{ñ}$ 而如果 $\epsilon ：\mathrm{中号}\to \mathbb{[R}$ 是一个正连续半代数函数，那么有一个 ${\mathcal{C}}^{\infty }$ 半代数函数 $G：\mathrm{中号}\to \mathbb{[R}$ 这样 $|F-G|<\epsilon$。我们证明了该结果到全局亚分析类别的一般化。我们的定理实际上适用于更大的框架，因为它适用于在多项式有界的o最小结构（扩展实数域）中定义的每个函数，${\mathcal{C}}^{\infty }$细胞分解。我们还为Lipschitz和${\mathcal{C}}^{\mathrm{1个}}$ 可定义的功能。