A local version of the Pawłucki-Pleśniak extension operator (Q1763778)

Let \(K\) be a perfect compact set on the line \(\mathbb{R}\) and let \(E(K)\) denote the space of all functions \(f\) on \(K\) that are extendable to some \(F\in C^\infty(\mathbb{R})\). Various methods have been devised for constructing continuous extension operator \(L\) such that \(L(f)= F\). One such method described by \textit{W. Plešniak} [J. Approximation Theory 61, 106--117 (1990; Zbl 0702.41023)] used Lagrange interpolation and required that \(K\) satisfy Markov's inequality. In the present paper, the authors construct \(L\) using local Lagrange interpolation on certain generalized Cantor-type sets \(K\) which do not require Markov's inequality. They rely upon earlier work of \textit{A. Goncharov} [Stud. Math. 126, No. 2, 161--170 (1997; Zbl 0911.46016)].