Sulla non validità di un teorema di approssimazione

Acquistapace, Francesca ; Broglia, Fabrizio ; Tognoli, Alberto

Atti della Accademia nazionale dei Lincei. Rendiconti della Classe di scienze fisiche, matematiche e naturali, Série 8, Tome 55 (1973) no. 3-4, pp. 207-209

Résumé

The following theorem is true: if $U$ is open in $\mathbf{R}^{n}$, $V \subset U$ is a coherent real analytic set, and $f : U \to \mathbf{R}$ is a $C^{\infty}$ function such that $f\mid_{V}$ is analytic, then it is possible to approximate $f$ (together with its derivatives) by analytic functions $\{ f_{n} \}$ such that $f_{n}\mid_{V} = f \mid_{V}$. In this paper we prove that this result is not true unless $V$ is coherent (with the reduced structure).