On polynomial and barycentric interpolations

The present survey collects some recent results on barycentric interpolation showing the similarity to the corresponding Lagrange (or polynomial) theorems. Namely we state that the order of the Lebesgue constant for barycentric interpolation is at least log n; we state a Grünwald–Marcinkiewicz type theorem for the barycentric case; moreover we define a Bernstein type process for the barycentric interpolation which is convergent for any continuous function. As far as we know this is the first process of this type. The analogue results for the polynomial Lagrange interpolation are well known.