Convergence of subdiagonal Pad\'{e} approximations of $C_{0}$-semigroups

TL;DR

This paper proves strong convergence of subdiagonal Padé approximations to $C_{0}$-semigroups on Banach spaces, providing explicit rates and applications to Laplace transform inversion.

Contribution

It establishes convergence of subdiagonal Padé approximations for $C_{0}$-semigroups, including explicit rates and results for special classes of semigroups.

Findings

Strong convergence on $ extrm{D}(A^{rac{1}{2}+eta})$ for $eta>rac{1}{2}$

Explicit convergence rates in $n$

Applications to Laplace transform inversion

Abstract

Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Pad\'{e} approximations of the exponential function. We prove that for $-A$ the generator of a uniformly bounded $C_{0}$-semigroup $T$ on a Banach space $X$, the sequence $(r_{n}(-tA))_{n \in\mathbb{N}}$ converges strongly to $T(t)$ on $\textrm{D}(A^{\alpha})$ for $\alpha>\frac{1}{2}$. Local uniform convergence in $t$ and explicit convergence rates in $n$ are established. For specific classes of semigroups, such as bounded analytic or exponentially $\gamma$-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.