The approximation of functions on arcs with zero angles
Let L be a Jordan arc on the complex plane \({\mathbb{C}}\). Let \(H^{\omega}(L)\) be the class of functions f defined on L such that \[ | f(z_ 1)-f(z_ 2)| \leq C\omega (| z_ 1-z_ 2|)\quad (z_ 1,z_ 2\in L), \] where \(\omega\) (\(\delta)\) \((\delta >0)\) is a modulus of continuity. Let h be the conformal mapping of \({\mathbb{C}}-L\) onto \(\{| z| >1\}\) with \(h(\infty)=\infty\). Put \[ L_ n:=\{z;\quad | h(z)| =1+1/n\},\quad \delta_ n=:\sup \{dist(z,L);\quad z\in \overline{int L_ n}\}. \] The author defines classes of arcs L which are characterized by the following condition: For each function f in \(H^{\omega}(L)\) there exists a sequence of polynomials \(\{P_ n\}\) such that \[ \deg P_ n\leq n,\quad \| f-P_ n\|_ L\leq C_ 1\omega (\delta_ n)\quad and\quad \| P_ n\|_ L\leq C_ 2\omega (\delta_ n)/\delta_ n\quad (n\geq 1), \] where \(C_ 1\), \(C_ 2\) are positive constants.
scientific article; zbMATH DE number 3973272