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O-minimal version of Whitney’s extension theorem. (English) Zbl 1318.14052
In the paper under review the authors give a generalized and improved version of [K. Kurdyka and W. Pawłucki, Stud. Math. 124, No. 3, 269–280 (1997; Zbl 0955.32006)]. The main result is the following:
Given an o-minimal structure on a real closed field \(R\), let \(\Omega\) be an open definable subset of \(R^n\) and let \(E\) be a definable closed subset of \(\Omega\). Let \(p,q\) be natural numbers with \(p\leq q\) and let \[ F(x,X)=\sum_{|\kappa|\leq p}\frac{1}{\kappa !}F^\kappa(x)X^\kappa \] be a definable \(C^p\)-Whitney field on \(E\) (i.e. all \(F^\kappa\)’s are definable \(C^p\)-functions). Then there exists a definable \(C^p\)-function \(f:\Omega\to R\) that is \(C^q\) on \(\Omega\setminus E\) such that \(D^\kappa f=F^\kappa\) on \(E\) for all \(\kappa\) with \(|\kappa|\leq p\).

MSC:
14P10 Semialgebraic sets and related spaces
32B20 Semi-analytic sets, subanalytic sets, and generalizations
03C64 Model theory of ordered structures; o-minimality
14P15 Real-analytic and semi-analytic sets
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