Polynomial approximation on three-dimensional real-analytic submanifolds of \(\mathbf{C}^n\) (Q2719015)

In this note the problem of approximating arbitrary continuous functions on a compact subset \(K\) of \(n\)-dimensional complex Euclidean space \(\mathbb{C}^n\) by polynomials in the coordinate functions \(z_1,\dots, z_n\) is considered. Let \(C(K)\) denote the space of all continuous complex-valued functions on \(K\), with norm \(\|g\|_K=\max\{|g(z)|:z\in K\}\), and let \(P(K)\) denote the closure of the set of polynomials in \(C(K)\). NEWLINENEWLINENEWLINEThe purpose of this paper is to prove the following result on approximation on three-manifolds:NEWLINENEWLINENEWLINETheorem. Let \(\Sigma\) be a real-analytic three-dimensional submanifold of \(\mathbb{C}^n\). Let \(K\) be a compact subset of \(\Sigma\) such that \(\partial K\) (the boundary of \(K\) relative to \(\Sigma\)) is a two-dimensional submanifold of class \(C^1\). If \(K\) satisfies the following two conditions: NEWLINENEWLINENEWLINE(i) \(K=\widehat{K}:=\{z\in\mathbb{C}^n:|Q(z)|\leq\|Q\|_K\) for every polynomial \(Q\}\), NEWLINENEWLINENEWLINE(ii) for each \(p\in K\), there exists \(g\in P(K)\) with \(g(p)=1\) and \(|g|<1\) on \(K\backslash\{p\}\),NEWLINENEWLINENEWLINEthen \(P(K)=C(K)\).