  We create a new, functional calculus, approach to approximation of
C_0-semigroups on Banach spaces. As an application of this approach, we obtain
optimal convergence rates in classical approximation formulas for
C_0-semigroups. In fact, our methods allow one to derive a number of similar
formulas and equip them with sharp convergence rates. As a byproduct, we prove
a new interpolation principle leading to efficient norm estimates in the Banach
algebra of Laplace transforms of bounded measures on the semi-axis.
