Two problems on interpolation (Q1909584)

The authors pose two open problems concerning polynomial interpolation, provide some information on related known results, and give conjectures for the answers. The two problems are the following, where \(\Delta_n\) is a set of \(n\) distinct interpolation points in the interval \([0, 1]\). Firstly, given a complex-valued continuous function on \([0, 1]\), it is possible to find sets \(\Delta_n\), such that the resulting interpolation polynomials of degree \(n\) converge uniformly to the given function. Secondly, is a Newton type interpolation procedure always possible, i.e. given a real-valued continuous function on \([0, 1]\), are there point sets with \(\Delta_n \subset \delta_{n+1}\), such that the corresponding interpolation polynomials converge uniformly to the function.