{"paper":{"title":"Convergence of subdiagonal Pad\\'{e} approximations of $C_{0}$-semigroups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NA","math.OA"],"primary_cat":"math.FA","authors_text":"Jan Rozendaal, Moritz Egert","submitted_at":"2012-10-31T17:33:27Z","abstract_excerpt":"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 i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.8408","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}