Padé-type approximants and linear functional transformations

Abstract

Let f(.)=$\mathop \Sigma \limits_{i = o}^\infty$ c_ig_i (.) be a series of functions and let F(.)=$\mathop \Sigma \limits_{i = o}^\infty$ c_ih_i (.) be the series obtained by applying a linear functional transformation to f. It is shown that the Padé-type approximants of F can be deduced from that of f by application of the same functional transform. Some examples and applications are given. Convergence theorems are obtained. The particular case of the Laplace transform is studied in more detail.