Convergence of subdiagonal Padé approximations of $C_{0}$-semigroups
This work provides rigorous convergence guarantees for a class of numerical approximations of semigroups, which is relevant for researchers in operator theory and numerical analysis.
The paper proves that subdiagonal Padé approximations of the exponential function converge strongly to uniformly bounded C0-semigroups on Banach spaces, with explicit convergence rates and local uniform convergence in time. Applications to vector-valued Laplace transform inversion are provided.
Let $(r_{n})_{n \in \mathbb{N}}$ be the sequence of subdiagonal Padé 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 $\textrm{D}(A^α)$ for $α>\frac{1}{2}$. Local uniform convergence in $t$ and explicit convergence rates in $n$ are established. For specific classes of semigroups, such as bounded analytic or exponentially $γ$-stable ones, stronger estimates are proved. Finally, applications to the inversion of the vector-valued Laplace transform are given.