Rational approximation of operator semigroups via the $\mathcal B$-calculus

TL;DR

This work strengthens the rational approximation theory for $C_0$-semigroups on Hilbert spaces by leveraging the refined $\\mathcal{B}$-calculus, producing sharper stability bounds and optimal rates that depend on the regularity of the initial data. The authors provide a self-contained construction of the $\\mathcal{B}$-calculus and demonstrate its superiority over the HP-calculus in yielding operator-norm estimates for rational approximants, including subdiagonal Padé schemes. They establish precise $\\mathcal{B}$-norm bounds for powers of stable rational functions and for the associated approximation errors, translating these into sharp convergence rates in $X$ for data in $\\mathrm{dom}(A^s)$. A notable result is the improved rates for subdiagonal Padé approximations, with explicit constants and a logarithmic correction whose sharpness is analyzed. Overall, the paper advances numerical semigroup analysis and PDE discretizations by providing tighter, data-dependent convergence guarantees via a robust functional-calculus framework.

Abstract

We improve the classical results by Brenner and Thomée on rational approximations of operator semigroups. In the setting of Hilbert spaces, we introduce a finer regularity scale for initial data, provide sharper stability estimates, and obtain optimal approximation rates. Moreover, we strengthen a result due to Egert-Rozendaal on subdiagonal Padé approximations of operator semigroups. Our approach is direct and based on the theory of the $\mathcal B$- functional calculus developed recently. On the way, we elaborate a new and simple approach to construction of the $\mathcal B$-calculus thus making the paper essentially self-contai

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3