  M. J. Beltr\'{a}n-Meneua et al. \cite{beltran1} and E. Jord\'{a} and A.
Rodr\'{i}guez-Arenas \cite{jorda} characterized the (uniformly) mean ergodic
composition operators on $H^\infty(\mathbb{D})$ and $H^\infty_\nu
(\mathbb{D})$, respectively. In this paper, by using the interpolating
sequences, we give other necessary and sufficient conditions for the
(uniformly) mean ergodicity of composition operators on these spaces.
