Some unusual results on extrapolation methods

This paper is devoted to properties of sequence transformations and the corresponding recursive algorithms for their implementation, which were never considered before. We first give necessary conditions that are satisfied if the transformed sequence converges faster than the initial one. These conditions can be used for deciding if a method is worth to be used. They also serve as the basis for defining criteria for stopping the acceleration algorithm when the best possible precision is obtained. Then, prescribing the transformed sequence, we show how to obtain the initial sequence which produces it via the transformation or via its recursive algorithm. These results show that almost any behavior is possible for the transformed sequence. A similar problem about Pade-type approximants is studied.