Barycentric rational interpolation with no poles and high rates of approximation

It is well known that rational interpolation sometimes gives better approximations than polynomial interpolation, especially for large sequences of points, but it is difficult to control the occurrence of poles. In this paper we propose and study a family of barycentric rational interpolants that have no poles and arbitrarily high approximation orders, regardless of the distribution of the points. The family includes a construction of Berrut as a special case. Keywords: Rational interpolation, polynomial interpolation, blending functions, approximation order.