Characterization of classes of ultradifferentiable functions that admit an analogue of the Whitney extension theorem. (Q1432224)

The main result announced in this article completes the characterization of surjectivity of the restriction map in the context of ultradifferentiable functions of Beurling or Roumieu type. The full proof appeared in [Math. Ann. 320, No. 1, 115--126 (2001; Zbl 0989.46019)]. Let \(\omega\) be a non-quasianalytic weight function in the sense of \textit{R. W. Braun, R. G. Meise} and \textit{B. A. Taylor} [Result. Math. 17, No. 3--4, 206--237 (1990; Zbl 0735.46022)]. If \(K\) is an arbitrary non-empty compact subset of \(\mathbb{R}^N\), we denote by \(E_{*}(K)\) the linear space of Whitney jets of Beurling or Roumieu type on \(K\). We denote by \(\rho_K\) the restriction map from the space \(E_{*}(\mathbb{R}^N)\) of all ultradifferentiable functions of Beurling or Roumieu type on \(\mathbb{R}^N\) into \(E_{*}(K)\). The following conditions are equivalent: (1) \(\rho_K\) is surjective for every non-empty compact subset \(K\); (2) \(\rho_K\) is surjective for one non-empty compact subset \(K\); (3) the weight \(\omega\) is strong. The strong weights were introduced by \textit{R. G. Meise} and \textit{B. A. Taylor} [Ark. Mat. 26, No. 2, 265--287 (1988; Zbl 0683.46020)]. \textit{R. W. Braun, R. G. Meise, B. A. Taylor} and the reviewer [Stud. Math. 99, No. 2, 155--184 (1991; Zbl 0738.46009)] showed that (3) implies (1). The fact that (2) implies (3) in the case that \(K\) is a singleton or a convex compact set with non-empty interior had already been proved by Meise and Taylor in the Beurling case and by them and the reviewer in the Roumieu case. The author proves this implication in the general case. The main step in the proof is the construction of a certain family of polynomials.