On the kernel of vector epsilon-algorithm and related topics

The vector epsilon-algorithm of Wynn is a powerful method for accelerating the convergence of vector sequences. Its kernel is the set of sequences which are transformed into constant sequences whose terms are their limits or antilimits. In 1971, a sufficient condition characterizing sequences in this kernel was given by McLeod. In this paper, we prove that such a condition is not necessary. Moreover, using Clifford algebra, we give a formula for the vector \epsilon_2-transformation, which is formally the same as in the scalar case, up to operations in a Clifford algebra. Hence, Aitken's \Delta_2 process is extended in this way to vectors. Then, we derive the explicit algebraic and geometric expressions of sequences of the kernel of the \epsilon_2-transformation. We also formulate a conjecture concerning the explicit algebraic expression of kernel of the vector epsilon-algorithm.