Approximation in the mean by Lagrange interpolation polynomials in the complex plane
Let the domain \(G\subset {\mathbb{C}}\) be bounded by a Jordan curve \(\gamma\), where \(\gamma\) has a parametric representation \(\Gamma\) with \(\Gamma '\in Lip(\alpha)\), \(\alpha >0\). Let f be analytic in G, continuous in \(\bar G\) and we denote by \(L_ n(f;z)\) the Lagrange interpolation polynomial associated to f and the \((n+1)th\) Fejér points. The author proves for \(0<p<\infty\) \[ (\int_{\gamma}| L_ n(f;z)-f(z)|^ p| dz|)^{1/p}\leq C\cdot \omega (f;n^{-1}), \] where C is a constant independent of f and n, \(\omega\) (f;\(\delta)\) is the modulus of continuity of f on \(\bar G.\) This result was earlier obtained by \textit{S. Ya. Al'per} and \textit{G. I. Kalinogorskaya} [Izv. Vyssh. Uchebn. Zaved. Mat. 1969, No.11(90), 13-23 (1969; Zbl 0215.422)] under stronger assumptions on \(\gamma\).
scientific article; zbMATH DE number 4122301