Approximation of Operator Semigroups Using Linear-Fractional Operator Functions and Weighted Averages

An analytic semigroup of operators on a Banach space is approximated by a sequence of positive integer powers of a linear-fractional operator function. It is proved that the order of the approximation error in the domain of the generating operator equals $O(n^{-2}\ln(n))$. For a self-adjoint positive definite operator $A$ decomposed into a sum of self-adjoint positive definite operators, an approximation of the semigroup {$\exp(-tA)$} ($t\geq0$) by weighted averages is also considered. It is proved that the order of the approximation error in the operator norm equals $O(n^{-1/2}\ln(n))$.