Best approximation and saturation on domains bounded by curves of bounded rotation (Q1806142)

Let \(G\) be a Jordan domain with a boundary curve of bounded rotation. By \(A(\overline G)\) is denoted the class of functions that are holomorphic in \(G\) and continuous on the closure \(\overline G\). Let further \(p_m(\overline G,z)\) be the Faber polynomial of degree \(m\) for the set \(\overline G\). For functions \(f\in A(\overline G)\) having the so-called Faber derivative \(\varphi =F^rf,r\in N\cup\{0\}\), satisfying the Hőlder condition of order \(\alpha ,0<\alpha\leq 1,\) and for a matrix \(\mu =(\mu _m^n)\), the folloving means are considered \[ U_n^{\mu} (f,\overline G,z)=a_0(f,\overline G)+\sum _{m=1}^{n}(im)^{-r}a_{m} (\varphi ,\overline G)\mu _m^np_m(\overline G,z),\quad z\in \mathbb{C}. \] Here \(a_m\) are the Faber coefficients of the function in the parentheses. Under certain restrictions on the matrix \(\mu \) a relation is proved which contains full information about the order of saturation. The applications of this result relative to the partial sums, Fej\(\acute e\)r, Riesz and Rogosinski means are given and the saturation order of these methods are established.