Extension operators with analytic values
By a classical theorem of Whitney, any Whitney jet \(f\) defined on a closed set \(F\subset \mathbb{R}^n\) can be extended to a function \(\widetilde{f}\in C^\infty( \mathbb{R}^n)\) which is real analytic on \( \mathbb{R}^n\setminus F\). The question if such an extension is possible by means of a continuous linear operator has recently been studied by several authors (see \textit{J. Schmets} and \textit{M. Valdivia} [Bull. Polish Acad. Sci. Math. 45, 359--367 (1997; Zbl 0902.26014)], \textit{M. Langenbruch} [Result. Math. 36, 281--296 (1999; Zbl 0945.26029)], \textit{L. Frerick} and \textit{D. Vogt} [Proc. Am. Math. Soc. 130, 1775--1777 (2002; Zbl 1007.46029)] and \textit{R. Brück} and \textit{L. Frerick} [Result. Math. 43, 56--73 (2003; Zbl 1043.46022)]). The authors improve these results by showing the following theorem which is optimal in the case of one variable: Assume that a continuous linear extension operator exists for the Whitney jets defined on \(F\subset \mathbb{R}\). Then the extension operator can be chosen such that the extensions are holomorphic functions on \( \mathbb{C}\setminus \mathbb{R}\) if and only if \(\partial F\) is compact. Moreover, if \(F\) is compact, the extensions can be chosen to be holomorphic on \( (\mathbb{C}\cup \{\infty\})\setminus \mathbb{R}\).
scientific article; zbMATH DE number 2067307