On theorems of Jackson and Bernstein type in the complex plane
The paper is concerned with the question: For which Jordan domains D are the generalizations of Jackson's and Bernstein's theorems valid? Let (J) denote the proposition: If \(f\in Lip \alpha\) in \(\bar D\) \((0<\alpha <1)\), then \(E_ n(f)=O(n^{-\alpha})\), and let (B) denote the converse proposition. Also, let \(\Phi\) be a conformal map from \(\{\) \(w: | w| >1\}\) onto \({\mathbb{C}}\setminus \bar D\) with \(\Phi (\infty)=\infty\), and let \(\Psi =\Phi^{-1}\). We then have the following four statements: (1) \(\Phi\in Lip 1\Rightarrow\) (J); (2) (J) \(\Rightarrow\Phi\in Lip 1\); (3) \(\Psi\in Lip 1\Rightarrow\) (B); (4) (B) \(\Rightarrow\Psi\in Lip 1.\) Of these, (3) was proved by Lebedev and Tamrazov in 1970, and (1) is true under slight additional assumptions on \(\Phi\) (Kövari, Belyi). Here it is shown that (2) and (4) are true under slight additional assumptions on \(\Phi\) or \(\Psi\). The authors conjecture that (1)-(4) are unconditionally true. - There are interesting side results: A local version of Bernstein's inequality, and results on the conformal mapping of strip domains.
scientific article; zbMATH DE number 4042255
scientific article