  In this paper we introduce vector-valued Hermite expansions to approximate
one-parameter operator families such as $C_0$-groups and cosine functions. In
both cases we estimate the rate of convergence of these Hermite expansions to
the related family and compare with other known approximations. Finally we
illustrate our results with particular examples of $C_0$-groups and cosine
functions and their Hermite expansions.
