Interpolation of individual functions (Q1904170)

The paper is dealing with some convergence problems concerning interpolation projections of functions in \(C[0,1]\) onto the subspace \({\mathcal P}_n\) of polynomials of degree \(n-1\). If \(f,g \in C[0,1]\) are such that \(f/g= p/q\), \(p,q \in {\mathcal P}_n\), then there exists a sequence \(\Delta_n\) of interpolation points such that the sequences of Lagrange interpolation polynomials \(L (\Delta_n)f\) and \(L (\Delta_n)g\) converge to \(f,g\) respectively (Cor. 3). If \(X\) is a finite union of closed disjoint intervals in \([0,1]\) and \(f\in C(X)\) then there exists a sequence \(\Delta_n \subset X\) of interpolation points such that \(L(\Delta_n) f \to f\) in \(C(X)\). Some negative results and open problems are included.