The divergence of the barycentric Pade approximants
This resolves a theoretical question about the convergence of a recently proposed approximation method, showing it inherits the divergence issues of classical Padé approximants.
The paper proves that barycentric Padé approximants, like standard Padé approximants, can diverge: for any polynomial P, there exists a power series S with arbitrarily small coefficients such that the approximants of P+S fail to converge uniformly on any domain with non-empty interior.
We explain that, like the usual Padé approximants, the barycentric Padé approximants proposed recently by Brezinski and Redivo-Zaglia can diverge. More precisely, we show that for every polynomial P there exists a power series S, with arbitrarily small coefficients, such that the sequence of barycentric Padé approximants of P + S do not converge uniformly in any subset of the complex plane with a non-empty interior.