On rates in Euler's formula for C_0-semigroups

TL;DR

This paper establishes optimal convergence rates for Euler's approximation of C_0-semigroups using functional calculus, covering various cases previously studied and demonstrating the limits of possible improvements.

Contribution

It introduces a method to determine optimal convergence rates for Euler's formula in the context of C_0-semigroups using functional calculus techniques.

Findings

Optimal convergence rates are achieved for Euler's approximation of C_0-semigroups.

The results encompass many previously studied special cases.

The convergence rates obtained are essentially the best possible.

Abstract

By functional calculus methods, we obtain optimal convergence rates in Euler's approximation formula for C_0-semigroups restricted to ranges of generalized Stieltjes functions. Our results include a number of partial cases studied in the literature and cannot essentially be improved.