Polynomial approximation in the \(L^p\)-metric on disjoint segments (Q1405666)

Let \(E=\bigcup_{k=1}^m [a_k,b_k]\) be a finite union of disjoined compact segments on the complex plane \(\mathbb C\). Let \(G\) be the Green function of the complement of \(E\) with pole at \(\infty \). Put \(\rho_h:=\{ z\in \mathbb C\setminus E; G(z)= \log(1+h)\}\), \(h>0\). Main result: If \(f\in \mathcal C(E)\) and \(f'\in L^{p_k}([a_k,b_k])\), \(k=1,\dots,m\), then there exists a sequence of polynomials \(P_n\) such that \(\deg P_n \leq n\) and \[ \sum_{k=1}^m \int_{[a_k,b_k]} \left| \frac {f(z)-P_n (z)}{\rho_{ 1/n} }\right | ^{p_k} | dz| \leq c, \quad n\geq 1, \] where \(c=c(E,f)\) is a positive constant that does not depend on \(n\). The class of continuous functions \(f\) on [-1,1] such that \(f'\in L^p([-1,1]), \quad 1<p<\infty\), was earlier described in terms of the rate of polynomial approximation in the \(L^p\)-metric by \textit{V. P. Motornyi} [Math. USSR, Izv. 5 (1971), 889--914 (1972; Zbl 0249.41003)], and \textit{M. K. Potapov} [Tr. Mat. Inst. Akad. Nauk Steklova 134, 260--277 (1975; Zbl 0326.41020)]. The construction of the approximating polynomials given by the present author is new even in the case of one segment.