Generalizations of Aitken's process for accelerating the convergence of sequences

When a sequence or an iterative process is slowly converging, a convergence acceleration process has to be used. It consists in transforming the slowly converging sequence into a new one which, under some assumptions, converges faster to the same limit. In this paper, new scalar sequence transformations having a kernel (the set of sequences transformed into a constant sequence) generalizing the kernel of the Aitken's $\Delta^2$ process are constructed. Then, these transformations are extended to vector sequences. They also lead to new fixed point methods which are studied.