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• W3105654089 abstract "In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:Sto T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:Sto T$ that are differentiable of class~${mathcal C}^nu$ for a fixed integer $nugeq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $nu=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $nu geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${mathcal C}^nu$ 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 ${mathcal C}^1$-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." @default.
• W3105654089 created "2020-11-23" @default.
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• W3105654089 date "2019-07-23" @default.
• W3105654089 modified "2023-10-02" @default.
• W3105654089 title "Differentiable approximation of continuous semialgebraic maps" @default.
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• W3105654089 doi "https://doi.org/10.1007/s00029-019-0496-5" @default.
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