The problem of approximation in mean on arcs in the complex plane (Q325522)

Suppose that \(\Gamma\) is a rectifiable Jordan curve with diameter \(d\) and let \(E^{(p)}_n(f,\Gamma)\) be the best approximation of a function \(f\,:\;\Gamma\to \mathbb{C}\) by algebraic polynomials of order at most \(n\) in the space \(L^p(\Gamma)\). For \(t\in\Gamma\), \(\delta >0\), denote by \(\theta_t(\delta)\) the Lebesgue measure of \(\Gamma_t(\delta):=\{\tau\in\Gamma\,:\;|t - \tau |\leq \delta\}\). A curve \(\Gamma\) belongs to the class \(S_{\theta}\) if there exists a constant \(C(\Gamma)\geq 1\) such that \(\sup_{t\in\Gamma}\theta_t(\delta)\leq C(\Gamma)\,\delta\) for all \(0 < \delta \leq d\). It is proved that if \(\Gamma\in S_{\theta}\) and \(p>2\), then for \(E^{(p)}_n(f,\Gamma)\) the Jackson estimate is true for all \(f\in L^p(\Gamma)\) (Theorem 1) and the integral Hölder class \(H^{\alpha}_{p,\Gamma}\), \(0<\alpha<1\), can be characterized by the condition \(E^{(p)}_n(f,\Gamma)\leq \text{const}\cdot n^{-\alpha}\) (Theorem 3).