On convergence rates in approximation theory for operator semigroups

TL;DR

This paper introduces a novel functional calculus approach to analyze the convergence rates of C_0-semigroup approximations on Banach spaces, achieving optimal and sharp estimates for classical formulas.

Contribution

It develops a new functional calculus method that yields optimal convergence rates and a new interpolation principle for Laplace transforms in Banach algebras.

Findings

Derived sharp convergence rates for classical approximation formulas.

Established a new interpolation principle for norm estimates.

Enabled derivation of similar formulas with precise convergence behavior.

Abstract

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.