Description of classes of functions with given rate of decrease of their best uniform polynomial approximations
Let E be a connected compact set in \({\mathbb{C}}\) such that \(D={\mathbb{C}}\setminus E\) is connected and each point of the Carathéodory compactification of D is a simple complex number. Let A(E) denote the set of functions, which are continuous on E and holomorphic in \(E^ 0\), and let \(E_ n(f,E)\) denote the best uniform approximation of f on E by polynomials of degree less or equal to n. If \(f\in A(E)\), \(z\in bE\), \(t>0\), \(k\geq 0\) (k is an integer), \(U(z,t)=\{v:\) \(| z-v| \leq t\}\), then the quantity \(\omega_{k,E}(f,z,t)=E_ k(f,E,U(z,t))\) is called a local modulus of continuity of f at z. Let d(t) be a nondecreasing, positive function on [0,\(\infty)\) such that \(d(0)=0\), \(d(st)\leq cs^ ad(t)\), where c and a are positive constants independent on s and t. Then, under some additional restrictions on E the following theorem holds. Theorem. The inequality \(E_ n(f,E)\leq const.d(1/n)\), \(n=1,2,..\). holds if and only if there exists a \(k\geq 0\) such that for each \(z\in bE\) and \(t>0\), \(\omega_{k,E}(f,z,t)\leq const.d(r(z,t))\), where the number \(r(z,t)>0\) is determined from the condition \(\inf_{v}| v-z| =t\), \(v\in L_{1+r(z,t)}\). Here \(L_{1+r}\) is the level line \(L_{1+r}=\{z:\) \(| F(z)| =1+r\}\).
scientific article; zbMATH DE number 3901129