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 \(O(n^{-1/2}\ln(n))\).