Factorization of certain evolution operators using lie algebra: Convergence theorems

This work concerns convergence theorems with a sequence of ξ-approximants for exponential evolution operators with Lie operator arguments. A companion paper presents the formulation of the ξ-approximants. The theorems presented in this paper give the conditions which are sufficient for convergence of the sequences. Although the main emphasis will be on convergence properties of the one-dimensional case, the generalization to multidimensional cases is quite straightforward.