Jackson-Bernstein theorem in strictly pseudoconvex domains in \({\mathbb{C}}^ n\)
Let \(\Omega\) be a bounded strictly pseudoconvex domain of \({\mathbb{C}}^ n\) with \(C^ 2\)-boundary. Denote by A(\(\Omega)\) the space of holomorphic function in the interior of \(\Omega\) and continuous on \({\bar \Omega}\) and by \(\Lambda^{\alpha}(\Omega)\), \(\alpha\in (0,1)\) the Hölder class of holomorphic functions on \({\bar \Omega}\). The main result: If the set of polynomials is dense in A(\(\Omega)\), then for any \(f\in \Lambda^{\alpha}(\Omega)\) there is a sequence of polynomials \(p_ N\), deg \(p_ N\leq N\) such that: \(| f(z)-p_ N(z)| <c_ f N^{-\alpha},\) \(z\in \partial \Omega.\) In the case \(\partial^{\beta}f\in \Lambda^{\alpha}(\Omega)\), for every multi-index \(\beta\) with \(| \beta |\) fixed, the author obtains analogous results.
scientific article; zbMATH DE number 4118707