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Rational approximation of holomorphic semigroups revisited

Charles Batty, Alexander Gomilko, Yuri Tomilov

Abstract

Using a recently developed $\mathcal H$-calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature.

Rational approximation of holomorphic semigroups revisited

Abstract

Using a recently developed -calculus we propose a unified approach to the study of rational approximations of holomorphic semigroups on Banach spaces. We provide unified and simple proofs to a number of basic results on semigroup approximations and substantially improve some of them. We show that many of our estimates are essentially optimal, thus complementing the existing literature.
Paper Structure (10 sections, 16 theorems, 164 equations)

This paper contains 10 sections, 16 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. The HP-calculus and the $\mathcal{H}$-calculus
  3. Estimates for rational approximations of the exponential
  4. Stability estimates
  5. Approximation rate estimates
  6. Stability of operator approximations
  7. Rates for operator approximations
  8. Optimality in terms of the algebra $\mathcal{H}_\psi$
  9. Optimality for semigroup generators
  10. Appendix: Proof of Lemma \ref{['asymptotic']}

Key Result

Theorem 1.1

Let $r$ be an $\mathcal{A}(\psi)$-stable rational approximation of order $q$ to the exponential, and let $A \in \operatorname{Sect}(\varphi)$ for some $\varphi\in [0,\theta)$ and $\theta \in (0,\psi)$. Then there exists a constant $C = C(\theta)$ such that the following hold for all $s\in [0,q]$, $n

Theorems & Definitions (38)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • proof
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Theorem 3.6
  • proof
  • ...and 28 more