Holomorphic extensions of Whitney jets
For a closed subset \(F\) of \(\mathbb{R}\), denote by \({\mathcal E}^\infty(F)\) the space of all Whitney jets of infinite order on \(F\), let \(F^\times:= (\mathbb{R}\setminus F)\times i\mathbb{R}\) and denote by \({\mathcal H}(G)\) the space of all functions that are holomorphic on the open set \(G\) in \(\mathbb{C}\). Furthermore, let \[ {\mathcal B}^\infty(\mathbb{R}):= \Bigl\{f\in{\mathcal C}^\infty(\mathbb{R}): \sup_{x\in\mathbb{R}}| f^{(j)}(x)|< \infty\text{ for each } j\in\mathbb{N}_0\Bigr\}. \] The authors show that in various extension results for Whitney functions on closed subsets of \(\mathbb{R}\) the domain of analyticity of possible extensions is larger than it was known so far. Their main results are combined in the following statement: Theorem. Let \(K\subset\mathbb{R}\) be compact. For each \((\varphi_j)^\infty_{j=0}\) in \({\mathcal E}^\infty(K)\) there exists \(f\in {\mathcal B}^\infty(\mathbb{R})\cap{\mathcal H}(K^\times)\) such that \(f^{(j)}|_K= \varphi_j\), \(j\in\mathbb{N}_0\). If there exists an extension operator \(E:{\mathcal E}^\infty(K)\to{\mathcal C}^\infty(\mathbb{R})\), then there also exists an extension operator \(\widehat E:{\mathcal E}^\infty(K)\to {\mathcal B}^\infty(\mathbb{R})\cap{\mathcal H}(K^\times)\). If \(K\) is only closed, then the above holds if \({\mathcal B}^\infty(\mathbb{R})\) is replaced by \(C^\infty(\mathbb{R})\). For \(K= [0,1]\), there is even an extension operator \(\widehat E: {\mathcal E}^\infty([0,1])\to{\mathcal H}(\widehat C\setminus [0,1])\).
scientific article; zbMATH DE number 2002340