Constructive theory of analytic functions on a quasidisk (Q1848423)

Let \(K\subset\mathbb C\) be a compact set with connected complement. Denote by \(\mathcal A(K)\) the class of all functions continuous on \(K\) and analytic in the interior of \(K\). Let \(\mathcal P_n\) be the class of complex polynomials of degree at most \(n\in\mathbb N\). For \(f\in\mathcal A(K)\) and \(n\in\mathbb N\), denote by \(E_n(f,K) :=\inf\{\|f-p\|_K\), \(p\in\mathcal P_n\} \) the best uniform approximations of \(f\) on \(K\). The following two problems are fundamental for the constructive theory of functions of a complex variable: (1) describe the rate of polynomial approximation of all functions \(f\in\mathcal A(K)\) with given smoothness properties; (2) describe all functions \(f\in\mathcal A(K)\) with given rate of decrease of \(E_n(f,K)\) as \(n\to\infty\). In this paper, the author considers the case of functions on a closed quasidisk \(\overline G\), i.e., a domain bounded by a quasiconformal curve \(\partial G\). Direct and inverse theorems, establishing a connection between the rate of polynomial approximation of the function \(f\) on the boundary \(\partial G\) and its smoothness properties, are proved. The main idea of these theorems is that both problems can be treated in the same way.